Fuzzy multiplier, sum and intersection rules in non-Lipschitzian settings: decoupling approach revisited

We revisit the decoupling approach widely used (often intuitively) in nonlinear analysis and optimization and initially formalized about a quarter of a century ago by Borwein&Zhu, Borwein&Ioffe and Lassonde. It allows one to streamline proofs of necessary optimality conditions and calculus relations, unify and simplify the respective statements, clarify and in many cases weaken the assumptions. In this paper we study weaker concepts of quasiuniform infimum, quasiuniform lower semicontinuity and quasiuniform minimum, putting them into the context of the general theory developed by the aforementioned authors. On the way, we unify the terminology and notation and fill in some gaps in the general theory. We establish rather general primal and dual necessary conditions characterizing quasiuniform $\varepsilon$-minima of the sum of two functions. The obtained fuzzy multiplier rules are formulated in general Banach spaces in terms of Clarke subdifferentials and in Asplund spaces in terms of Fr\'echet subdifferentials. The mentioned fuzzy multiplier rules naturally lead to certain fuzzy subdifferential calculus results. An application from sparse optimal control illustrates applicability of the obtained findings.


Introduction
When dealing with problems involving several component functions or sets, e.g., proving necessary optimality conditions in metric or normed spaces or establishing subdifferential/normal cone calculus relations in normed spaces, it is common to consider extensions Czech Academy of Sciences, Institute of Mathematics, Žitná 25, 115 67 Prague 1, Czech Republic, E-mail: fabian@math.cas.cz,ORCID: 0000-0003-3031-0862 of the problems allowing the components to depend on or involve individual variables while ensuring that these individual variables are not too far apart.This decoupling method (the term coined by Borwein and Zhu [8]) allows one to express the resulting conditions in terms of subdifferentials of the individual functions and/or normal cones to individual sets, or appropriate primal space tools.
For instance, when dealing with the problem of minimizing the sum of two extendedreal-valued functions ϕ 1 and ϕ 2 , one often replaces the function of a single variable x → (ϕ 1 + ϕ 2 )(x) with the function of two variables (decoupled sum [31]) (x 1 , x 2 ) → ϕ 1 (x 1 ) + ϕ 2 (x 2 ) while forcing the distance d(x 1 , x 2 ) to be small.The latter is often done by adding a penalty term (or an increasing sequence of terms) containing d(x 1 , x 2 ).Here and throughout the paper, for brevity, we restrict ourselves to the case of two functions.The definitions and statements can be easily extended to an arbitrary finite number of functions.
The decoupling approach has been intuitively used in numerous publications for decades.As claimed in [8,Section 6.1.4],all basic subdifferential rules in Banach spaces are different facets of a variational principle in conjunction with a decoupling method.The basics of the decoupling method were formalized in a series of publications by Borwein and Zhu [7,8], Borwein and Ioffe [6], Lassonde [31] and Penot [38].With regards to the mentioned above minimization problem, the following uniform infimum [31] Λ U (ϕ 1 , ϕ 2 ) := lim inf of (ϕ 1 , ϕ 2 ) over (or around) U plays a key role.Here U is a given set.It can represent a set of constraints or a neighbourhood of a given point.Observe that, thanks to d(x 1 , x 2 ) → 0, condition dist (x 1 ,U) → 0 is equivalent to dist (x 2 ,U) → 0; hence definition (1.1) gives no advantage to the variable x 1 .The quantity Λ U (ϕ 1 , ϕ 2 ) from (1.1) is referred to in [8] as decoupled infimum and in [38] as stabilized infimum.As pointed out in [8,31], it is involved in many conditions associated with decoupling methods in nonlinear analysis and optimization.The earlier publications [6,7] employ also a simplified version of (1.1): (ϕ 1 (x 1 ) + ϕ 2 (x 2 )). (1.2) As shown in Proposition 3.1 (vi), definitions (1.1) and (1.2) are not too different, especially in the situation of our main interest in the current paper when U represents a neighbourhood of a point in many situations.It follow directly from definitions (1.1) and (1.2) that and the inequalities can be strict (see Example 3.3).The requirements that Λ U (ϕ 1 , ϕ 2 ) or Λ • U (ϕ 1 , ϕ 2 ) coincide with the conventional infimum of ϕ 1 + ϕ 2 represent important qualifi- cation conditions.
A more restrictive sequential definition of uniform lower semicontinuity ((ULC) condition) was introduced in [6, Definition 6] (see also [8, Definition 3.3.17]): (ϕ 1 , ϕ 2 ) is sequen- tially uniformly lower semicontinuous (or coherent [38,Lemma 1.124]) on U if, for any sequences {x 1k }, {x 2k } ⊂ U satisfying d(x 1k , x 2k ) → 0 as k → +∞, there exists a sequence This definition was formulated in [6,8] for the case when U is a ball in a Banach space, but is meaningful in our more general setting, too.At the same time, one needs to be a little more careful to ensure that the expression under the lim sup in (1.4b) is well defined.It suffices to assume that {x 1k } ⊂ dom ϕ 1 and {x 2k } ⊂ dom ϕ 2 .The key point that distinguishes this definition from the one in the previous paragraph is the presence of conditions (1.4a), which relate the variable of ϕ 1 + ϕ 2 with those of the decoupled sum (x 1 , x 2 ) → ϕ 1 (x 1 ) + ϕ 2 (x 2 ).
Recall that the minimizing sequences involved in the expressions compared in (1.3) (see (1.2)) are entirely independent.As observed in [8,Section 3.3.8](see also Proposition 4.4), sequential uniform lower semicontinuity possesses certain stability which makes it more convenient in applications.
Thanks to Proposition 4.3 (iii) the sequential uniform lower semicontinuity property admits an equivalent analytical representation.We call it firm uniform lower semicontinuity; see Definition 4.1 (iii).
Employing the decoupled definitions (1.1) and (1.2) and the respective associated concepts of (firm) uniform lower semicontinuity and local uniform minimum allows one to streamline proofs of optimality conditions and calculus relations, unify and simplify the respective statements, as well as clarify and in many cases weaken the assumptions.For instance, it was emphasized in [6, Remark 2] that (ULC) condition (firm uniform lower semicontinuity in the language adopted in the current paper) covers the three types of situations in which (strong) fuzzy calculus rules had been established for appropriate subdifferentials in Banach spaces earlier: when the underlying space is finite-dimensional, when one of the functions has compact level sets and when all but one functions are uniformly continuous.
Among the fuzzy calculus rules the following (strong) fuzzy sum rule is central: For any x ∈ dom ϕ 1 ∩ dom ϕ 2 , x * ∈ ∂ (ϕ 1 + ϕ 2 )( x) and ε > 0, there exist points x 1 , x 2 such that Here, ∂ usually stands for the Fréchet subdifferential.This type of rules have been established in appropriate spaces also for other subdifferentials; see [6][7][8]31].Note that none of the aforementioned three types of situations involves the traditional (for this type of results) assumption that all but one functions are locally Lipschitz continuous, thus, ruining the widely spread (even now) myth that Lipschitzness is absolutely necessary, at least, in infinite-dimensional spaces.
The fact that in finite dimensions the above fuzzy sum rule is valid for arbitrary lower semicontinuous functions has been known since the mid-1980s; see [18,Theorem 2].A similar result is true also for weakly lower semicontinuous functions in Hilbert spaces; it is usually formulated in terms of proximal subdifferentials; see [11,Theorem 1.8.3].By means of an example, it has been shown in [42] that the Hilbert space fuzzy sum rule fails if the weak sequential lower semicontinuity is replaced by just lower semicontinuity.Both the finite-dimensional and Hilbert space fuzzy sum rules can be proved without using the Ekeland variational principle.In more general spaces some additional assumptions are required like compactness of the level sets of one of the functions or uniform (but not necessarily Lipschitz) continuity of all but one functions.The decoupling approach formalized in [6-8, 31, 38] allows one to treat all these situations within the same framework.Note that, unlike the finite-dimensional case, in infinite dimensions without additional assumptions strong fuzzy sum rules may fail; see a counterexample in [42,Theorem 1].For Fréchet subdifferentials, even with the mentioned additional assumptions such a rule is only valid in Asplund spaces, and this property is characteristic for Asplund spaces; see [35].
In contrast to the sum rule above, the so-called weak fuzzy sum rule is valid for lower semicontinuous functions in (appropriate) infinite-dimensional spaces without additional assumptions; see [7,8,17,19,38].Instead of condition (1.6b) involving the distance, it employs the condition where U * is an arbitrary weak* neighbourhood of zero in the dual space.The weak fuzzy sum rule immediately yields the validity of the conventional (strong) fuzzy sum rule in finite dimensions.
In our recent paper [28] the decoupling approach was used intuitively when proving the main result [28,Theorem 4.1].When analyzing later the proof of that theorem and related definitions and facts, and tracing the ideas back to the foundations in [6][7][8]31], we have realized that the 'novel notions of semicontinuity' discussed in [28,Section 3] are closely related to the uniform lower semicontinuity as in [6,7,31].More importantly, our version of uniform lower semicontinuity is actually weaker, thus, leading to weaker notions of uniform infimum, firm uniform infimum and local uniform minimum as well as fuzzy optimality conditions and calculus relations under weaker assumptions.Further developing the notions introduced and studied in [28] and putting them into the context of the general theory developed in [6-8, 31, 38] is the main aim of the current paper.On the way, we unify the terminology and notation, and fill in some gaps in the general theory.
We clearly distinguish between the uniform lower semicontinuity defined by (1.3) and the firm uniform lower semicontinuity (the analytical counterpart of the sequential lower semicontinuity defined using (1.4); see Definition 4.1 (iii)) exposing the advantages of the latter stronger property.The novel weaker properties arising from [28] are called quasiuniform lower semicontinuity and firm quasiuniform lower semicontinuity.The first one is defined similarly to (1.3) using instead of (1.2) the quasiuniform infimum Here EI(U) stands for the collection of all essentially interior subsets of U; see Definition 2.1.Clearly, , and the inequality can be strict; see Examples 3.2 and 3.3.To simplify the comparison, all four uniform lower semicontinuity notions together with their localized (near a point) versions are collected in a single Definition 4.1.
We study the weaker than (1.5) local quasiuniform minimality notion: employing the quasiuniform infimum (1.8), together with the related notions of quasiuniform stationarity and quasiuniform ε-minimality; see Definition 6.1.Using these new notions allows one to formulate more subtle conditions.The mentioned quasiuniform minimality/stationarity coincide with the corresponding conventional local minimality, stationarity and ε-minimality when the pair (ϕ 1 , ϕ 2 ) is quasiuniformly lower semicontinuous on an appropriate neighbourhood of x.We establish rather general primal and dual (fuzzy multiplier rules) necessary conditions characterizing quasiuniform ε-minimum of the sum of two functions.Under the assumption of quasiuniform lower semicontinuity of (ϕ 1 , ϕ 2 ), they characterize the conventional ε-minimum and, as a consequence, also any stationary point and local minimum.The sufficient conditions for quasiuniform lower semicontinuity discussed in the paper encompass all known conditions ensuring fuzzy multiplier rules.These are formulated in general Banach spaces in terms of Clarke subdifferentials and in Asplund spaces in terms of Fréchet subdifferentials.In general Banach spaces Clarke subdifferentials can be replaced in this type of statements by the G-subdifferentials of Ioffe [20].The mentioned fuzzy multiplier rules naturally lead to certain fuzzy subdifferential calculus results under rather weak assumptions; see Theorem 7.1.The structure of the paper is as follows.The next Section 2 contains the basic notation and some preliminary results from variational analysis used throughout the paper.In particular, we introduce essentially interior subsets that are key for the definition of quasiuniform infimum and other new notions, and can be useful elsewhere.In Section 3 we discuss the notions of uniform and quasiuniform infimum and two other 'decoupling quantities' as well as several analogues of the qualification condition (1.3).Diverse examples illustrate the computation of the 'decoupling quantities'.These 'decoupling quantities' and qualification conditions provide the basis for the definitions of uniform and quasiuniform lower semicontinuity and their 'firm' analogues discussed in Section 4. We show that firm uniform and firm quasiuniform lower semicontinuity properties are stable under uniformly continuous perturbations of the involved functions and prove several sufficient conditions for the mentioned uniform lower semicontinuity properties.In Section 5 we investigate the situation where at least one of the involved functions is the indicator function of a set and discuss the notions of relative uniform and quasiuniform lower semicontinuity.We show that the situations when a pair of indicator functions are not firmly uniformly or firmly quasiuniformly lower semicontinuous are rare.The firm quasiuniform lower semicontinuity of a pair of indicator functions near a point in the intersection of the sets is implied, for instance, by the well-known and widely used subtransversality property.Section 6 is devoted to the problem of minimizing the sum of two functions.Here we prove rather general primal and dual necessary conditions characterizing quasiuniform ε-minimum and formulate several consequences.In Section 7 we illustrate the value of quasiuniform lower semicontinuity in the context of subdifferential calculus.An application in sparse optimal control is considered in Section 8.The paper closes with some concluding remarks in Section 9.

Notation and preliminaries
Basic notation and definitions Our basic notation is standard; cf.e.g.[20,24,35].Throughout the paper X and Y are either metric or normed spaces (or more specifically Banach or Asplund spaces).We use the same notation d(•, •) and • for distances and norms in all spaces (possibly with a subscript specifying the space).Normed spaces are often treated as metric spaces with the distance determined by the norm.If X is a normed space, x ∈ X and U ⊂ X, we use x +U := U +x := {x +u | u ∈ U} for brevity of notation.The topological dual of X is denoted by X * , while •, • denotes the bilinear form defining the pairing between the two spaces.If not stated otherwise, product spaces are equipped with the associated maximum distances or norms.The associated dual norm is the sum norm.In a metric space B and B are the open and closed unit balls, while B δ (x) and B δ (x) are the open and closed balls with radius δ > 0 and centre x, respectively.We write B * and B * to denote open and closed unit balls in the dual to a normed space.
The distance from a point x to a set Ω is defined by dist (x,U) := inf x∈U d(x, u) with the convention dist (x, / 0) := +∞.Furthermore, for two sets In this paper intU and clU represent the interior and the closure of U, respectively.In a normed space we use coU and cl coU to denote the convex hull and the closed convex hull of U, respectively.We write x k → x to denote the (strong) convergence of a sequence {x k } to a point x.In a normed space, x k ⇀ x expresses the weak convergence of {x k } to x, i.e., x * , x k → x * , x for each x * ∈ X * .Similarly, x * k * ⇀ x * denotes the weak* convergence of {x * k } to x * in the dual space.Symbols R, R + and N represent the sets of all real numbers, all nonnegative real numbers and all positive integers, respectively.We make use of the notation R ∞ := R∪{+∞} and the conventions inf / 0 R = +∞ and sup / 0 R + = 0, where / 0 (possibly with a subscript) denotes the empty subset (of a given set).Definition 2.1 Let X be a metric space and We write EI(U) and EI cl (U) to denote the collections of, respectively, all essentially interior subsets and all closed essentially interior subsets of U.
The next lemma summarizes basic properties of essentially interior subsets of a given set.
Lemma 2.1 Let X be a metric space and U ⊂ X.The following assertions hold: Suppose X is a normed space.
Proof Most of the assertions are direct consequences of Definition 2.1.

coV ∈ EI(U).
⊓ ⊔ Remark 2.1 Assertion (ix) of Lemma 2.1 may fail if X is merely a metric space.Indeed, let X be the closed interval [0, 2] in R with the conventional distance, x := 1 and V := {0}.It is easy to see that B 2 (V ) ⊂ B 1 ( x) = X; hence V ∈ EI(B 1 ( x)).However, V ⊂ B ρ ( x) for any ρ ∈ (0, 1).Note that in this example For an extended-real-valued function f : X → R ∞ , its domain and epigraph are defined, respectively, by dom f The next lemma provides a connection between stationarity and ε-minimality.
Lemma 2.2 Let X be a metric space, f : X → R ∞ and x ∈ dom f .Then x is stationary for f if and only if for any ε > 0, there is a δ ε > 0 such that, for any δ ∈ (0, δ ε ), x is an εδ -minimum of f on B δ ( x).
⊓ ⊔ For a set-valued mapping F : X ⇒ Y , its domain and graph are defined respectively, by dom Recall that a Banach space is Asplund if every continuous convex function on an open convex set is Fréchet differentiable on a dense subset [39], or equivalently, if the dual of each separable subspace is separable.A Banach space is Fréchet smooth if it has an equivalent norm that is Fréchet differentiable away from zero [8,26,29].All reflexive, particularly, all finite-dimensional Banach spaces are Fréchet smooth, while all Fréchet smooth spaces are Asplund.We refer the reader to [8,15,35,39] for discussions about and characterizations of Asplund and Fréchet smooth spaces.

Subdifferentials, normal cones and coderivatives
Below we review some standard notions of generalized differentiation which can be found in many monographs; see, e.g., [10,20,35].Below X and Y are normed spaces.
For a function ϕ : X → R ∞ and a point x ∈ dom ϕ, the (possibly empty) set is the Fréchet subdifferential of ϕ at x.If x is a local minimum (or, more generally, a stationary point) of ϕ, then obviously 0 ∈ ∂ ϕ( x) (Fermat rule).If X is Asplund, the limiting subdifferential of ϕ at x can be defined as the outer/upper limit (with respect to the norm topology in X and weak* topology in X * ) of Fréchet subdifferentials: For a subset Ω ⊂ X and a point x ∈ Ω , the closed convex (possibly trivial) cone The Clarke normal cone to Ω at x is defined by where is the Clarke tangent cone to Ω at x, while the Clarke subdifferential of a function ϕ : X → R ∞ at x ∈ dom ϕ can be defined via (the direct definition of the Clarke subdifferential is a little more involved and employs Clarke-Rockafellar directional derivatives).It always holds , and whenever Ω and ϕ are convex, the above normal cones and subdifferentials reduce to the conventional constructions of convex analysis: For a mapping F : X ⇒ Y between normed spaces and a point ( x, ȳ) ∈ gph F, the setvalued mapping D * F( x, ȳ) : is the Fréchet coderivative of F at ( x, ȳ).If F is single-valued and ȳ = F( x), we write simply D * F( x) for brevity.
Preliminary results Here we recall some fundamental results from the literature used in the sequel.We start with the celebrated Ekeland variational principle; see e.g.[4,8,20,24,35].Lemma 2.3 Let X be a complete metric space, ϕ : X → R ∞ lower semicontinuous and bounded from below and x ∈ dom ϕ.Then, for any ε > 0, there exists a point x ∈ X satisfying the following conditions: The next lemma summarizes some standard sum rules for Fréchet and Clarke subdifferentials as well as conventional subdifferentials of convex functions which can be found in many monographs on variational analysis [8,10,20,21,39,40,44].Lemma 2.4 Let X be a Banach space, ϕ 1 , ϕ 2 : X → R ∞ and x ∈ dom ϕ 1 ∩ dom ϕ 2 .Then the following assertions hold.
(i) Differentiable sum rule [26]. 39, 44].If ϕ 1 and ϕ 2 are convex and ϕ 1 is continuous at a point in dom ϕ 2 , then ∂ (ϕ [10,41].If ϕ 1 is Lipschitz continuous near x and ϕ 2 is lower semicon- tinuous near x, then (iv) Fuzzy sum rule [16, 19].If X is Asplund, ϕ 1 is Lipschitz continuous near x and ϕ 2 is lower semicontinuous near x, then, for any x * ∈ ∂ (ϕ 1 + ϕ 2 )( x) and ε > 0, there exist points x 1 , x 2 ∈ X such that conditions (1.6) hold true.(v) Weak fuzzy sum rule [8,19,38].If X is Fréchet smooth and ϕ 1 and ϕ 2 are lower semi- continuous near x, then, for any x * ∈ ∂ (ϕ 1 + ϕ 2 )( x), ε > 0 and a weak* neighbourhood U * of zero in X * , there exist x 1 , x 2 ∈ X such that conditions (1.6a) and (1.7) hold true.Remark 2.2 (i) The sum rules in parts (iii) and (iv) of Lemma 2.4 contain the standard (and commonly believed to be absolutely necessary) assumption of Lipschitz continuity of one of the functions.In fact, this assumption is not necessary.For the fuzzy sum rule in part (iv), it has been shown in [13,Corollary 3.4 (ii)] that it suffices to assume one of the functions to be uniformly continuous in a neighbourhood of the reference point.In the setting of smooth spaces the latter fact was discussed also in [7,8,31,38].(ii) Part (v) of Lemma 2.4 shows, in particular, that the fuzzy sum rule holds inherently in finite dimensions without assuming one of the functions to be Lipschitz continuous near the reference point, thus, strengthening the assertion in part (iv) of Lemma 2.4.(iii) The sum rules in parts (i), (ii) and (iii) of Lemma 2.4 are exact in the sense that the subdifferentials (and the derivative in part (i)) in their right-hand sides are computed at the reference point.In contrast, the rules for Fréchet subdifferentials in parts (iv) and (v) are often referred to as fuzzy or approximate because the subdifferentials in the righthand sides of the inclusions are computed at some other points arbitrarily close to the reference point.

Uniform and quasiuniform infimum
In this section we discuss the notions of uniform and quasiuniform infimum and two other 'decoupling quantities' as well as several analogues of the qualification condition Thus, inf U (ϕ 1 + ϕ 2 ) < +∞.Recall that the uniform infimum of (ϕ 1 , ϕ 2 ) over (or around) U is defined by either (1.1) or (1.2), while the quasiuniform infimum Λ † U (ϕ 1 , ϕ 2 ) of (ϕ 1 , ϕ 2 ) over U is defined by (1.8).Some elementary properties of these quantities are collected in the next proposition.
(vi).It follows from (1.1) and ( 1. Combining both estimates gives (vi).(vii).Denote by M the expression in the first representation.Then, by (1.8), we have This proves the first representation.The other two repre- sentations follow from definition (1.8) and the first representation thanks to Lemma 2.1 (v).Indeed, if any of these restrictions is violated, then ϕ 1 (x 1 ) + ϕ 2 (x 2 ) = +∞, and, in view of the convention inf / 0 R = +∞, such points do not affect the value of the lim inf.(iii) The restriction x 1 ∈ V in the first and third representations in part (vii) of Proposition 3.1 can be replaced with x 2 ∈ V .Analogous replacements can be made in the second and fourth representations in part (viii).
The inequalities in parts (i) and (iv) of Proposition 3.1 can be strict.Inequality (1.3), opposite to the second inequality in part (i), is an important qualification condition.We are going to show that in some important situations it can be replaced by a weaker (thanks to the first inequality in part (iv) and Example 3.3) condition inf Note that, unlike (1.3), inequality (3.2) can be strict; see Example 3.2.The quantities compared in (1.3) or (3.2) are computed independently.At the same time, it is important in some applications to ensure that, given an appropriate sequence of (x 1 , x 2 ) with d(x 1 , x 2 ) → 0 as in (1.2) or (1.8), the corresponding x approximating the infimum in the left-hand side can be chosen close to x 1 and x 2 (which are close to each other because d(x 1 , x 2 ) → 0).To accommodate for this additional requirement, we are going to utilize the following definitions: with the notation Thanks to assumption (3.1) the latter expression is well defined as long as x 1 ∈ dom ϕ 1 and Some elementary properties of the quantities (3.3) and equivalent representations of the quantity (3.3b) are collected in the next proposition. ) (iii) The following estimates are true: (iv) The following representations hold true: (v) If X is a normed space, x ∈ X and δ > 0, then the following representations hold true: Proof The assertions are direct consequences of definitions from (3.3).For the first inequality in (i) in the case intU = / 0, recall the convention sup / 0 R + = 0.For the inequalities in (ii), observe from (3.4) that inf for all x 1 ∈ dom ϕ 1 and x 2 ∈ dom ϕ 2 .Assertion (iii) clearly follows as ϕ 1 (x 1 ) + ϕ 2 (x 2 ) < α for some α > 0 immediately gives x 1 ∈ dom ϕ 1 and x 2 ∈ dom ϕ 2 .For the representations in (iv) and (v), reuse the arguments in the proof of Proposition 3.1 (vii) and (viii).
⊓ ⊔ The assumptions in part (ii) of Proposition 3.2 ensure that the left-hand sides of (3.5) and (3.6) are well defined.Recall that inf The restriction x 1 ∈ V in the first and third representations in part (iv) of Proposition 3.2 can be replaced with x 2 ∈ V .Similar replacements can be made in the second and fourth representations in part (v).
Then there exists a sequence {x k } ⊂ X such that conditions (1.4) are satisfied.Since x 1k ∈ V , it follows from Definition 2.1 that x k ∈ U for all sufficiently large k ∈ N, and consequently, lim k→∞ (ϕ 2) is a consequence of Proposition 3.2 (ii) and Remark 3.3 (ii).Thanks to (i), we have In view of (i), the latter condition is equivalent to Let δ > 0. Thanks to (i) and definition (3.8), we have ) by assertions (v) and (viii) of Lemma 2.1, and consequently, This proves implications (c) ⇒ (f) ⇒ (e).Furthermore, V ⊂ B δ ( x) for any V ∈ EI(B δ ′ ( x)) and, thanks to definition (3.8) and (i), we have lim sup This proves implication (e) ⇒ (d).
3) and (3.2) are formulated as equalities.This is because of the presence of the terms d(x, x 1 ) and d(x, x 2 ) in definition (3.4), preventing (ϕ 1 ♦ϕ 2 ) U (x 1 , x 2 ) from being negative.Besides, they do not allow to separate in (3.4) the terms containing x 1 , x 2 on one hand and x on the other hand (as in (1.3)
Example 3.2 Let lower semicontinuous convex functions ϕ 1 , ϕ 2 : R → R be given by and Particularly, (3.5) holds as equality while inequality (3.6) is strict.⊓ ⊔ The next example illustrates a situation where, again, first inequality in Proposition 3.1 (iv) is strict while both (3.5) and (3.6) hold as equalities.

⊓ ⊔
The final example of this section shows that if condition (3.1) is violated, then in inequalities (1.3) and (3.2) both sides may equal +∞.It also provides a situation when the inequalities in Proposition 3.2 (iii) are strict.

Uniform and quasiuniform lower semicontinuity
In this section we discuss certain uniform and quasiuniform lower semicontinuity properties of pairs of functions resulting from definitions (1.2), (1.8) and (3.3) of uniform infima which turn out to be beneficial in the context of the fuzzy multiplier and calculus rules.
1 compare (some form of) the infimum of the sum x → (ϕ 1 + ϕ 2 )(x) with (some form of) the lower limit of the decoupled sum (x 1 , x 2 ) → ϕ 1 (x 1 ) + ϕ 2 (x 2 ) as d(x 1 , x 2 ) → 0. These conditions prevent the lower limit of the decou- pled sum from being smaller than the infimum of the sum.(iv) The properties in parts (i) and (ii) of Definition 4.1 are defined via inequalities (1.3) and (3.2), while the definition of their firm counterparts in parts (iii) and (iv) use equalities . This is because of the presence of the terms d(x, x 1 ) and d(x, x 2 ) in definition (3.4) which prevent (ϕ 1 ♦ϕ 2 ) U (x 1 , x 2 ) from being negative; see Remark 3.4.(v) Slightly weaker versions of the 'firm' properties can be considered, corresponding to replacing (i) uniformly lower semicontinuous on U if and only if, for any ε > 0, there exists an η > 0 such that, for any x 1 , x 2 ∈ U with d(x 1 , x 2 ) < η, there is an x ∈ U satisfying (ii) quasiuniformly lower semicontinuous on U if and only if, for any V ∈ EI(U) and ε > 0, there exists an η > 0 such that, for any x 1 ∈ V and x 2 ∈ X with d(x 1 , x 2 ) < η, there is an x ∈ U satisfying condition (4.2); (iii) firmly uniformly lower semicontinuous on U if and only if, for any ε > 0, there exists an η > 0 such that, for any x 1 ∈ dom ϕ 1 ∩U and x 2 ∈ dom ϕ 2 ∩U with d(x 1 , x 2 ) < η, there is an x ∈ U ∩ B ε (x 1 ) satisfying condition (4.2); (iv) firmly quasiuniformly lower semicontinuous on U if and only if, for any V ∈ EI(U) and ε > 0, there exists an η > 0 such that, for any x 1 ∈ dom ϕ 1 ∩ V and x 2 ∈ dom ϕ 2 with d(x 1 , x 2 ) < η, there is an x ∈ B ε (x 1 ) satisfying condition (4.2).
Thanks to Proposition 3.3 (iv), it is possible to state a simplified characterization of firm quasiuniform lower semicontinuity near a point.It replaces the closed ball B δ ( x) with the open ball B δ ( x) and requires condition Θ † B δ ( x) (ϕ 1 , ϕ 2 ) = 0 to hold not for all, but for some δ > 0.
The (iv) Another example of a seeming lack of symmetry appears in parts (ii) and (iv) of Proposition 4.1: they require x 1 to belong to some essentially interior subset V , while there are no such restrictions on x 2 .This is because they use equivalent representations of Θ † U (ϕ 1 , ϕ 2 ) from Proposition 3.2 (iv).Recall that definition (3.3b) of Θ † U (ϕ 1 , ϕ 2 ) requires both x 1 and x 2 to belong to V .Thus, these characterizations can be rewritten in a (formally slightly more restrictive) symmetric form.A similar observation applies to the characterization in Proposition 4.2.(v) If X is a normed space, and U = B δ ( x) or U = B δ ( x) for some x ∈ X and δ > 0, one can use the characterizations from Proposition 3.
We now show that firm uniform and firm quasiuniform lower semicontinuity properties are stable under uniformly continuous perturbations of the involved functions.Proposition 4.4 Suppose that (ϕ 1 , ϕ 2 ) is firmly uniformly (resp., firmly quasiuniformly) lower semicontinuous and g : X → R is uniformly continuous on U. Then (ϕ 1 , ϕ 2 + g) is firmly uniformly (resp., firmly quasiuniformly) lower semicontinuous on U.

⊓ ⊔
Let us point out that the proof of Proposition 4.4 heavily exploits the nature of firm uniform/quasiuniform lower semicontinuity.The assertion may not be true if it is replaced by its non-firm counterpart.
The next proposition collects several sufficient conditions for firm uniform lower semicontinuity.
Given any points The next proposition exploits compactness assumptions in order to guarantee the presence of the uniform lower semicontinuity properties from Definition 4.1.
When it comes down to minimization problems in infinite-dimensional spaces, weakly sequentially lower semicontinuous functions are of special interest.Proposition 4.7 Let X be a normed space, ϕ 1 and ϕ 2 be weakly sequentially lower semi- continuous on U and inf U ϕ 2 > −∞.The pair (ϕ 1 , ϕ 2 ) is (i) uniformly lower semicontinuous on U if {x ∈ U | ϕ 1 (x) ≤ c} is weakly sequentially com- pact for each c ∈ R; (ii) quasiuniformly lower semicontinuous on U if cl w V ⊂ U and {x ∈ cl w V | ϕ 1 (x) ≤ c} is weakly sequentially compact for each V ∈ EI(U) and c ∈ R, where cl w V stands for the weak closure of V , i.e., the closure of V with respect to the weak topology in X.
The proof of Proposition 4.7 basically repeats that of parts (i) and (ii) of Proposition 4.6, replacing strong convergence with the weak one, and we skip it.Note that weak convergence does not allow us to establish similar sufficient conditions for the 'firm' properties.

Remark 4.5
The assertion in part (ii) of Proposition 4.7 remains true if cl w V is replaced by the weak sequential closure of V .However, the weak sequential closure of a set does not need to be weakly sequentially closed in general, see e.g.[33] for a study addressing so-called decomposable sets in Lebesgue spaces, so the associated statement has to be used with care.Corollary 4.1 Let X be a reflexive Banach space, U be convex and bounded and ϕ 1 and ϕ 2 be weakly sequentially lower semicontinuous on U. Then the pair (ϕ 1 , ϕ 2 ) is quasiuniformly lower semicontinuous on U.
Proof Let V ∈ EI(U).By Lemma 2.1 (x), coV ∈ EI(U), and, by Lemma 2.1 (v), cl coV ∈ EI(U).Hence, cl w V ⊂ cl w coV = cl coV ⊂ U. Since cl w V is, particularly, weakly sequentially closed and bounded while the sublevel sets of weakly sequentially lower semicontinuous functions are weakly sequentially closed, the set {x ∈ cl w V | ϕ 1 (x) ≤ c} is weakly se- quentially compact for each c ∈ R as X is reflexive.The assertion follows from Proposition 4.7 (ii).

Relative uniform and quasiuniform lower semicontinuity
In this section we investigate the situation where at least one of the involved functions is the indicator function of a set.We start with the case ϕ 1 := ϕ and ϕ 2 := i Ω for some function ϕ : X → R ∞ on a metric space X and subset Ω ⊂ X.Our basic assumption (3.1) in this setting becomes where U is another subset of X. From (1.2), (3.3), (3.4), Proposition 3.1 (vii), Proposition 3.2 (iv) and Remark 3.2, we obtain: uniformly/quasiuniformly/firmly uniformly/firmly quasiuniformly lower semicontinuous relative to Ω near a point x ∈ dom ϕ ∩ Ω if it is uniformly/quasiuniformly/firmly uniformly/firmly quasiuniformly lower semicontinuous relative to Ω on B δ ( x) for all sufficiently small δ > 0. The characterizations of the relative lower semicontinuity properties in the next two propositions are consequences of representations (5.2) and corresponding assertions in Propositions 3.1 to 3.3.They can also be derived from Proposition 4.1.
Proposition 5.1 The function ϕ is (i) uniformly lower semicontinuous relative to Ω on U if and only if, for any ε > 0, there exists an η > 0 such that, for any x ∈ U with dist (x, Ω ∩ U) < η, there is a u ∈ Ω ∩ U such that ϕ(u) < ϕ(x) + ε; (ii) quasiuniformly lower semicontinuous relative to Ω on U if and only if, for any V ∈ EI(U) and ε > 0, there exists an η > 0 such that, for any x ∈ V with dist (x, Ω ) < η, there is a u ∈ Ω ∩U such that ϕ(u) < ϕ(x) + ε; (iii) firmly uniformly lower semicontinuous relative to Ω on U if and only if, for any ε > 0, there exists an η > 0 such that, for any x ∈ dom ϕ ∩U with dist (x, Ω ∩U) < η, there is a u ∈ Ω ∩U ∩ B ε (x) such that ϕ(u) < ϕ(x) + ε; (iv) firmly quasiuniformly lower semicontinuous relative to Ω on U if and only if, for any V ∈ EI(U) and ε > 0, there exists an η > 0 such that, for any x Remark 5.2 Proposition 5.1 (i) strengthens [38, Lemma 1.123 (a)]: if ϕ is uniformly lower semicontinuous around Ω [38, page 88], then it is uniformly lower semicontinuous relative to Ω on X.
The next proposition gives a simplified characterization of firm quasiuniform relative lower semicontinuity near a point.It is a consequence of Proposition 4.2.
Example 3.1 illustrates the firm uniform relative semicontinuity property (see Remark 4.2 (viii)).The following infinite-dimensional example will be important later on when we discuss applications of our findings.For fixed functions x a , x b ∈ L 2 (D) satisfying x a (ω) ≤ 0 ≤ x b (ω) for almost all ω ∈ D, we define the box-constraint set Ω ⊂ L 2 (D) by means of a.e. on D}, (5.4) and note that Ω is nonempty, closed and convex.For an x ∈ L 2 (D), we define A simple calculation shows that u x ∈ L 2 (D) is the uniquely determined projection of x onto Ω , and consequently, dist (x, Ω ) = x − u x .Furthermore, by construction, we have ϕ(u x ) ≤ ϕ(x).Given any x ∈ Ω , δ > 0, ε > 0 and η ∈ (0, ε), conditions x ∈ B δ ( x) and dist (x, Ω ) < η yield u x ∈ Ω ∩ B ε (x) and i.e., u x ∈ B δ ( x).By Remark 4.2 (v) and Proposition 5.1 (iii), ϕ is firmly uniformly lower semicontinuous relative to Ω near any point in Ω .Note that the function ϕ is discontinuous and not weakly sequentially lower semicontinuous.In fact, ϕ is nowhere Lipschitz continuous, see [34,Corollary 3.9].Clearly, i Ω is discontinuous.Thus, we have constructed a pair of uniformly lower semicontinuous functions, both non-Lipschitz, while one of them is not weakly sequentially lower semicontinuous.⊓ ⊔ Next, we discuss sufficient conditions for firm uniform and quasiuniform lower semicontinuity of a function relative to a set.For sufficient conditions for (not firm) quasiuniform lower semicontinuity of a function relative to a set, we refer the interested reader to [ (ii) firmly quasiuniformly lower semicontinuous relative to The next example illustrates the difference between uniform and quasiuniform relative lower semicontinuity.

⊓ ⊔
The case of two indicator functions of some subsets Ω 1 , Ω 2 ⊂ X can be consid- ered as a particular case of the uniform/quasiuniform lower semicontinuity properties in Definition 4.1 or relative lower semicontinuity properties in Definition 5.1.The corresponding properties are rather weak and are satisfied almost automatically in most natural situations. Let ) is automatically uniformly (and quasiuniformly) lower semicontinuous on U. Using (5.2c), (5.2d) and (5.2) as well as parts (i) and (iv) of Proposition 3.3, we can formulate characterizations of firm uniform and quasiuniform lower semicontinuity.
Proposition 5.5 The pair (i Ω 1 , i Ω 2 ) is (i) firmly uniformly lower semicontinuous on U if and only if lim sup (ii) firmly quasiuniformly lower semicontinuous on U if and only if (iii) firmly quasiuniformly lower semicontinuous near a point x ∈ Ω 1 ∩ Ω 2 if and only if for some δ > 0 it holds: The next proposition is a consequence of Proposition 5.4.The statement and its corollary show that the situations when a pair of indicator functions is not firmly uniformly or firmly quasiuniformly lower semicontinuous are rare.Proposition 5.6 Let Ω 1 and Ω 2 be closed.The pair (ii) firmly quasiuniformly lower semicontinuous on U if the sets Ω 1 ∩ clV are compact for all V ∈ EI(U).
Corollary 5.1 Let X be a finite dimensional Banach space, Ω 1 , Ω 2 be closed, and U be bounded.Then (i Ω 1 , i Ω 2 ) is firmly quasiuniformly lower semicontinuous on U.
The next statement gives alternative characterizations of the firm uniform and quasiuniform lower semicontinuity of a pair of indicator functions. ( (ii) firmly quasiuniformly lower semicontinuous on U if and only if (iii) firmly quasiuniformly lower semicontinuous near a point x ∈ Ω 1 ∩ Ω 2 if and only if for some δ > 0 it holds: Proof We prove the second assertion.The proofs of the first and the third ones follow the same pattern with some obvious simplifications.Observe that (5.8) trivially implies (5.6).Thanks to Proposition 5.5 (ii), it suffices to show the opposite implication.Let condition (5.6) be satisfied, and let Then for each k ∈ N, there exist points x 1k ∈ Ω 1 and , there exists a subset V ′ ∈ EI(U) such that V ∈ EI(V ′ ).Then x 1k ∈ V ′ for all sufficiently large k ∈ N. By (5.6), dist (x 1k , Ω 1 ∩ Ω 2 ) → 0, and consequently, dist (x k , Ω 1 ∩ Ω 2 ) → 0. Thus, condition (5.8) holds true.⊓ ⊔ Remark 5.3 In view of Proposition 5.7 (iii), the firm quasiuniform lower semicontinuity of a pair of indicator functions near a point in the intersection of the sets is implied, for instance, by the well known and widely used subtransversality property (also known as linear regularity, metric regularity, linear coherence and metric inequality), and as a consequence, also by the stronger transversality property (also known under various names); see, e.g., [9,20,27].
Recall that the sets Ω 1 and Ω 2 are subtransversal at x ∈ Ω 1 ∩ Ω 2 if there exist numbers α > 0 and δ > 0 such that Nonlocal versions of this property, i.e., with some subset U ⊂ X (e.g., U = X) in place of B δ ( x) are also in use.Condition (5.10) describes so called linear subtransversality.More subtle nonlinear, in particular, Hölder subtransversality (see e.g.[12]) is still sufficient for the property (5.9).
The following example, which is inspired by Example 3.1, shows that the firm uniform lower semicontinuity of a pair of indicator functions can be strictly weaker than (linear) subtransversality of the involved sets.

Optimality conditions
We consider here the problem of minimizing the sum of two functions ϕ 1 , ϕ 2 : X → R ∞ on a metric space X.When discussing dual optimality conditions X will be assumed Banach or, more specifically, Asplund.This model is quite general (see a discussion in [31]).It may represent so-called composite optimization problems, where typically the smoothness properties of ϕ 1 and ϕ 2 are rather different.If one of the functions is an indicator function, the model covers nonsmooth constrained optimization problems.As in the previous sections, we are going to exploit the decoupling approach, allowing ϕ 1 and ϕ 2 to take different inputs.We mostly discuss local minimality/stationarity properties of ϕ 1 + ϕ 2 around a given point x ∈ dom ϕ 1 ∩ dom ϕ 2 .Recall that x is called a local uniform minimum [31] of ϕ 1 + ϕ 2 if it satisfies (1.5).This notion is stronger than the conventional local minimum.Together with the related definitions of uniform infimum (1.1) and (1.2) and uniform lower semicontinuity (1.3) they form the foundations of the decoupling approach; see [6][7][8]31].In what follows, we examine weaker local quasiuniform minimality and stationarity concepts which are based on the decoupling quantity (1.8), and utilise the properties discussed in Section 3 as well as the quasiuniform lower semicontinuity from Section 4. Definition 6.1 (i) The point x is a local quasiuniform minimum of ϕ 1 + ϕ 2 if condition (1.9) is satisfied.If the latter condition is satisfied with δ = +∞, then x is referred to as a quasiuniform minimum of (iii) The point x is a quasiuniform stationary point of ϕ 1 + ϕ 2 if for any ε > 0, there exists a δ ε > 0 such that, for any δ ∈ (0, δ ε ), x is a quasiuniform εδ -minimum of (this is an immediate consequence of assertions (i), (iii) and (vi) of Proposition 3.1).Thanks to this observation, the assertion is a consequence of Proposition 3.1 (iv).

⊓ ⊔
The properties in Definition 6.1 imply the corresponding conventional local minimality/stationarity properties of ϕ 1 + ϕ 2 .As the following proposition reveals, they become equivalent when the pair (ϕ 1 , ϕ 2 ) is quasiuniformly lower semicontinuous (in the sense of Definition 4.1 (ii)) on an appropriate neighbourhood of x.
Suppose now that X is an Asplund space, and set ξ := 2ε/δ − ε > 0. By the fuzzy sum rule combined with the convex sum rule (see Lemma 2.4), applied to (6.12), there exist a point (x 1 , x 2 ) arbitrarily close to ( x1 , x2 ) with ϕ(x 1 , x 2 ) arbitrarily close to ϕ( x1 , x2 ) and a subgradient ) such that, taking into account (6.2a), (6.14c) and (6.14d), the following estimates hold true: and, in view of (6.13) and (6.14a), conditions (6.10a) and (6.11) are satisfied.⊓ ⊔ Remark 6.2 (i) Since the functions ϕ 1 and ϕ 2 in Theorems 6.1 and 6.2 are assumed to be lower semicontinuous, they are automatically bounded from below on some neighbourhood of x.We emphasize that Theorems 6.1 and 6.2 require x to be a quasiuniform ε-minimum of ϕ 1 + ϕ 2 , and ϕ 1 and ϕ 2 to be bounded from below on the same fixed neighbourhood of x. (ii) Theorem 6.2 generalizes and strengthens [28,Theorem 4.5 As consequences of Theorems 6.1 and 6.2 we obtain primal and dual necessary conditions for a local quasiuniform stationary point of a sum of functions.Corollary 6.1 Let X be complete and ϕ 1 , ϕ 2 be lower semicontinuous.Suppose that x is a quasiuniform stationary point of ϕ 1 + ϕ 2 .Then, for any ε > 0, there is a ρ ∈ (0, ε) such that, for any η > 0, there exist a number γ > 0 and points x1 , x2 ∈ X such that conditions (6.2a) are satisfied, and where the function ϕ γ : X → R is defined by (6.3).
Corollary 6.2 Let X be a Banach space and ϕ 1 , ϕ 2 be lower semicontinuous.Suppose that x is a quasiuniform stationary point of ϕ 1 + ϕ 2 .Then, for any ε > 0, there exist points x 1 , x 2 ∈ X such that conditions (1.6a) and (6.10b) are satisfied, and If X is Asplund, then, for any ε > 0, there exist points x 1 , x 2 ∈ X such that conditions (1.6a) are satisfied, and Below we provide a combined proof of the two corollaries.
Proof of Corollaries 6.1 and 6.2 Let ε > 0 and η := ε/2.By the assumptions, there exists a δ ∈ (0, ε) such that and x is a quasiuniform ηδ -minimum of ϕ 1 + ϕ 2 on B δ ( x).Thus, all the assumptions of Theorems 6.1 and 6.2 are satisfied with ε ′ := ηδ in place of ε.Observe that 2ε ′ /δ = ε, and almost all the conclusions follow immediately.We only need to show that, in the case of Corollary 6.2, ϕ i (x i ) − ϕ i ( x) < ε, i = 1, 2. Comparing conditions (6.10b) and (6.11a) in Theorem 6.2, we see that condition (6.11a) is valid in the general as well as in the Asplund space setting.By (6.11a) and (6.17) we have Let us revisit the setting in Example 3.1.

⊓ ⊔
As another consequence of Theorem 7.1 we can derive a fuzzy chain rule in a comparatively mild setting; cf.[26, Section 1.2] and [36,Section 3].It employs a firm relative quasiuniform lower semicontinuity qualification condition which holds trivially, e.g., if the involved outer function is uniformly continuous.The last condition obviously implies (7.2b).

⊓ ⊔ 8 An application in optimal control
We revisit the setting of Example 5.1.Let D ⊂ R d be some bounded open set.We consider a continuously differentiable mapping S from L 2 (D) to a Hilbert space H, the so-called control-to-observation operator, which assigns to each control function x ∈ L 2 (D) an observation S(x) ∈ H. Typically, S represents the composition of the solution operator associated with a given variational problem (e.g. a partial differential equation or a variational inequality) and some mapping which sends the output of the variational problem to the observation space H.In optimal control, a function x often has to be chosen such that S(x) is close to some desired object y d ∈ H which can be modeled by the minimization of the smooth term 1 2 S(x) − y d 2 .There are often other requirements which have to be respected in many situations.For example, a control has to belong to a simple constraint set Ω ⊂ L 2 (D) or has to be sparse, i.e., it has to vanish on large parts of the domain D. Here, we take a closer look at the sparsity-promoting function ϕ : L 2 (D) → R given in (5.3).Furthermore, we assume that Ω is given as in (5.4) where x a , x b ∈ L 2 (D) satisfy x a (ω) < 0 < x b (ω) almost everywhere on D. We note that Ω is closed and convex, so the various normal cones to this set coincide with the one in the sense of convex analysis.We investigate the optimal control problem min{ f (x) + ϕ(x) where f : L 2 (D) → R is an arbitrary continuously differentiable function and keep in mind that a possible choice for f would be the typical target-type function x → 1 2 S(x) − y d 2 + σ 2 x 2 where σ ≥ 0 is a regularization parameter.We identify the dual space of L 2 (D) with L 2 (D).Thus, for any x ∈ L 2 (D), f ′ (x) can be interpreted as a function from L 2 (D).Problems of type (OC) were already considered e.g. in [22,37,43] from the viewpoint of necessary and sufficient optimality conditions as well as numerical solution methods.Before we can state necessary optimality conditions for this optimization problem it has to be clarified how the subdifferentials of ϕ look like.This has been investigated in the recent paper [34].Before presenting the formulas we need to recall the concept of socalled slowly-decreasing functions, see [34, Definition 2.4, Theorem 2.10] as well as the discussions therein.In our first result we present approximate stationarity conditions for (OC).
Theorem 8.1 Let x ∈ L 2 (D) be a local minimum of (OC).Then, for each ε > 0, there exist a slowly decreasing function x 1 ∈ B ε ( x), some x 2 ∈ Ω ∩ B ε ( x) and x Now, the remaining assertions of the theorem follow from Lemma 8.1 and the well-known characterization of the normal cone N Ω (x 2 ).⊓ ⊔ We now take the limit as ε ↓ 0 in the system (8.1) in order to obtain a conventional stationarity condition.for almost every ω ∈ D. For almost every ω ∈ { x = x a }, we have x 1k (ω) < 0 as well as x 2k (ω) < 0 and, thus, x * k (ω) ≤ 0 for large enough k ∈ N, see (8.3b) as well, and taking the limit as k → +∞ gives (8.2b).Similarly, we can show (8.2c).Finally, for almost every ω ∈ { x = 0} ∩ {x a < x < x b }, we have ω ∈ {x 1k = 0} and ω ∈ {x a < x 2k < x b } for large enough k ∈ N, giving x * k (ω) = 0 for any such k ∈ N, and taking the limit gives (8.2a).⊓ ⊔

Example 5 . 1
Let D ⊂ R d be a Lebesgue-measurable set with positive and finite Lebesgue measure λ(D).We equip D with the σ -algebra of all Lebesgue-measurable subsets of D as well as (the associated restriction of) the Lebesgue measure λ, and consider the Lebesgue space L 2 (D) of all (equivalence classes of) measurable, square integrable functions equipped with the usual norm.In what follows, we suppress Lebesgue for brevity.Define a function ϕ :L 2 (D) → R by means of ∀x ∈ L 2 (D) : ϕ(x) := λ({x = 0}).(5.3)We use the notation {x = 0} := {ω ∈ D | x(ω) = 0} for brevity.Furthermore, the sets {x = 0}, {x < 0}, {x > 0} and analogous sets with non-vanishing right-hand side or bilateral bounds are defined similarly.We note that, by definition of L 2 (D), these sets are well defined up to subsets of measure zero.Particularly, ϕ from (5.3) is well defined.By means of Fatou's lemma one can easily check that ϕ is lower semicontinuous, see[34, Lemma 2.2].

1 . 5 . 3
28, Section 3.3].The next two statements are direct consequences of Propositions 4.5, 4.6 and 5.Proposition If ϕ is uniformly continuous on U, then it is firmly uniformly lower semicontinuous relative to Ω on U. Proposition 5.4 Let ϕ be lower semicontinuous on U and Ω be closed.The function ϕ is (i) firmly uniformly lower semicontinuous relative to Ω on U if {x ∈ U | ϕ(x) ≤ c} is compact for each c ∈ R, and lim sup x∈dom ϕ∩U,dist (x,Ω ∩U )→0 ϕ(x) < +∞;
next proposition gives sequential reformulations of the characterizations of quasiuniform lower semicontinuity properties from Proposition 4.1.The pair (ϕ 1 , ϕ 2 ) is (i) uniformly lower semicontinuous on U if and only if, for any sequences {x 1k } ⊂ dom ϕ 1 ∩ U and {x ii) Condition (4.2) automatically implies that x ∈ dom ϕ 1 ∩dom ϕ 2 .The condition obviously only needs to be checked for x 1 ∈ dom ϕ 1 and x 2 ∈ dom ϕ 2 .In view of Proposition 4.1, the properties in parts (ii) and (iv) of Definition 4.1 can only be meaningful when dom ϕ 1 ∩V = / 0 and dom ϕ 2 ∩V = / 0 for some V ∈ EI(U).(iii) Due to condition x ∈ B ε (x 1 ) involved in Proposition 4.1 (iv) and Proposition 4.2, it looks as if the point x 1 plays a special role in the firm quasiuniform lower semicontinuity property.In fact, both x 1 and x 2 contribute equally to this property (see definition (3.4) and Remark 3.2), and the mentioned condition can be replaced there with x ∈ B ε (x 1 ) ∩ B ε (x 2 ).
2k } ⊂ dom ϕ 2 ∩U satisfying d(x 1k , x 2k ) → 0 as k → +∞, there exists a sequence {x k } ⊂ U such that condition (1.4b) is satisfied; (ii) quasiuniformly lower semicontinuous on U if and only if, for any sequences {x 1k } ⊂ dom ϕ 1 ∩ U and {x 2k } ⊂ dom ϕ 2 satisfying {x 1k } ∈ EI(U) and d(x 1k , x 2k ) → 0 as k → +∞, there exists a sequence {x k } ⊂ U such that condition (1.4b) is satisfied; (iii) firmly uniformly lower semicontinuous on U if and only if, for any sequences {x 1k } ⊂ dom ϕ 1 ∩U and {x 2k } ⊂ dom ϕ 2 ∩U satisfying d(x 1k , x 2k ) → 0 as k → +∞, there exists a sequence {x k } ⊂ U such that conditions (1.4) are satisfied; (iv) firmly quasiuniformly lower semicontinuous on U if and only if, for any sequences {x 1k } ⊂ dom ϕ 1 ∩ U and {x 2k } ⊂ dom ϕ 2 satisfying {x 1k } ∈ EI(U) and d(x 1k , x 2k ) → 0 as k → +∞, there exists a sequence {x k } ⊂ X such that conditions (1.4) are satisfied.Remark 4.2 (i) Unlike the characterizations in the other parts of Propositions 4.1 and 4.3, those in parts (iv) of these statements do not require explicitly that x or x k belong to U or B δ ( x).However, in view of Proposition 3.3 (i), the mentioned conditions are automatically satisfied in these characterizations.( 1 (viii) to replace the arbitrary essentially interior subsets V in part (ii) of Proposition 4.1 by either the family of closed balls B ρ ( x) or the family of open balls B ρ ( x) with ρ ∈ (0, δ ).(vi) In view of Lemma 2.1 (v) collection EI(U) in parts (ii) and (iv) of Proposition 4.1 can be replaced with its sub-collection EI cl (U).(vii) In view of the characterization in Proposition 4.3 (iii) the (ULC) property [6, Definition 6] (also known as sequential uniform lower semicontinuity [8, Definition 3.3.17])at x is equivalent to the firm uniform lower semicontinuity on B δ ( x) for some δ > 0. Thanks to parts (i) and (iii) of Proposition 4.3, a pair of functions is uniformly lower semicontinuous (resp., firmly uniformly lower semicontinuous) on X if it is quasicoherent (resp., coherent) [38, Lemma 1.124].(viii) The properties in Definition 4.1 are rather weak.This is illustrated by the examples in Section 3.For pairs of functions ϕ 1 and ϕ 2 and corresponding sets U in Examples 3.1 to 3.4, it has been shown that inf U and condition (4.2) is satisfied withx := x 1 .(ii)Let sequences {x 1k } ⊂ dom ϕ 1 ∩ U and {x 2k } ⊂ dom ϕ 2 ∩ U satisfy d(x 1k , x 2k ) → 0 as k → +∞.Then x 2k = x.Thus, x 1k → x as k → +∞ and, in view of the lower semicontinuity of ϕ 1 , the conditions in Proposition 4.3 (iii) are satisfied with x k := x for all k ∈ N. (iii) This is a consequence of (i) and Proposition 4.4.
⊓ ⊔ Remark 4.3 (i) Condition (i) in Proposition 4.5 is satisfied if ϕ 2 is constanteverywhere or is the indicator function of a set containing dom ϕ 1 ∩ U. Function ϕ 1 in condition (i) does not have to be lower semicontinuous in the conventional sense.(ii) Proposition 4.5 with condition (iii) strengthens [7, Proposition 2.7.1] and is similar to [31, Proposition 2.1 (d)] and [8, Exercise 3.2.9](which use (1.1) instead of (1.2) in the qualification condition (1.3)).
If U is compact, then all the compactness assumptions as well as assumption inf U ϕ 2 > −∞ in Proposition 4.6 are satisfied automatically.(iii) As illustrated by Example 3.4 (see also Remark 4.2 (viii)), condition (4.4) in part (iv) of Proposition 4.6 is essential.(iv) Using slightly weaker versions of the 'firm' properties, corresponding to replacing ]. (iii) In view of Proposition 6.3 (ii) the conclusions of Theorems 6.1 and 6.2 are valid for the conventional ε-minimum if the functions are quasiuniformly lower semicontinuous on B δ ( x). (iv) The proof of the first (general Banach space) part of Theorem 6.2 uses the Clarke subdifferential sum rule (Lemma 2.4 (iii)).Clarke subdifferentials can be replaced in Theorem 6.2 by any subdifferentials possessing such an exact (see Remark 2.2 (iii)) sum rule in general Banach spaces.One can use for that purpose, e.g., the G-subdifferentials of Ioffe; see [20, Theorem 4.69].
The dual necessary conditions in Corollary 6.2 hold not necessarily at the reference point, but at some points arbitrarily close to it.That is why such conditions are referred to as approximate or fuzzy.Such conditions hold under very mild assumptions and also possess several interesting algorithmic applications; see e.g.[2, 3, 5-8, 14, 16-20, 25, 26,  28-32, 35]and the references therein.(iii)Condition(6.16)inCorollary 6.2 represents a rather standard Asplund space approximate multiplier rule (see e.g.[26, 29, 35]), while the general Banach space approximate multiplier rule (6.15) in terms of Clarke subdifferentials is less common.In fact, we do not know if it has been explicitly formulated in the literature.Note that Corollary 6.2 does not assume one of the functions to be locally Lipschitz continuous (or even uniformly continuous) as is common for multiplier rules in nonsmooth settings.(iv) The multiplier rules in Corollary 6.2 are deduced for a quasiuniform stationary point/local minimum, see also assertion (i).Thanks to Proposition 6.3 (i) and (iii), they apply to conventional stationary points/local minima when the pair of functions is quasiuniformly lower semicontinuous near the reference point.Several sufficient conditions ensuring this property are given in Propositions 4.5 to 4.7 and Corollary 4.1.In particular, the property holds if one of the functions is uniformly continuous (particularly if it is Lipschitz continuous) near the reference point.With this in mind, the second part of Corollary 6.2 generalizes the conventional Asplund space approximate multiplier rule and makes it applicable in more general situations.
[35,pproximate sum rule, similar to Theorem 7.1 with condition (i), is established in[8,  Theorem 3.3.19]underthestrongerassumption of firm uniform lower semicontinuity (in a Fréchet smooth space).The next immediate corollary of Theorem 7.1 gives a sufficient condition for the fuzzy intersection rule for Fréchet normals in reflexive Banach spaces.It employs no qualification conditions and improves the assertion of[35, Lemma 3.1].