Strongly regular points of mappings

In this paper, we use a robust lower directional derivative and provide some sufficient conditions to ensure the strong regularity of a given mapping at a certain point. Then, we discuss the Hoffman estimation and achieve some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the coefficient of the error bound.

In other words, the tangent manifold to f -1 (0) is equal tox + Ker f (x), where Ker f (x) is the set of those x such that f (x)(x) = 0.
We refer to Dontchev [2] for a nice overview on the Lyusternik theorem and to the fact that the Lyusternik theorem can be easily obtained from the Graves theorem. We also refer to the forthcoming book by Thibault [3]. Theorem 1.2 (Graves,[4]) Let X and Y be Banach spaces,x ∈ X, and f : X → Y be a C 1 -mapping whose derivative f (x) is onto. Then, there exist a neighborhood U ofx and a constant c > 0 such that for every x ∈ U and τ > 0 with B(x, τ ) ⊂ U, B(x, cτ ) ⊂ f B(x, τ ) (partial openness property with linear rate).
Ioffe and Tihomirov showed in [5] that the original Lyusternik proof may lead to a stronger result and proved that if f (x) is surjective, then there are κ > 0 and δ > 0 such that Ioffe's remark leads to a standard definition: Definition 1.1 Pointx ∈ X is said to be a regular point of a mapping f : X → Y if the relation (1.1) is satisfied.
In this note, we will callx a strongly regular point of f if the inequality holds locally, for all x belonging to a neighborhood ofx, where κ > 0 is a positive constant. Next, we will provide sufficient conditions forx to be a strongly regular point. Our results allow us to estimate the constant κ in (1.2). Then, we apply our results to the Hoffman estimate and obtain some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the upper limit of the error. In particular, for a finite-dimensional space X and a linear (continuous) mapping A : X → X, we prove that the estimate holds for all x ∈ X (Corollary 3.2 below). We can easily see that this estimate is sharp for injective linear mappings, in the sense that, if A is an injective linear mapping and inf A(u) : u ∈ X, dist(u, Ker A) = 1 < μ, Our work is outlined as follows. In Sect. 1, we recall the famous Lyusternik theorem and survey briefly its relationship with the concept of metric regularity. In Sect. 2, we first introduce the notion of homogeneous continuity of mappings. Then, using an appropriate notion of lower directional derivative, we achieve some results ensuring in finite dimension that for a given mapping a point is strongly regular. Finally, in Sect. 3, we focus our attention on Hoffman's estimate of approximate solutions of finite systems of linear inequalities and prove some similar estimates.

Sufficient conditions of regularity via generalized derivative
Throughout the paper, we use standard notations. For a normed space X, we denote its norm by · and by X * its (continuous) dual. The symbol S stands for the unit sphere, that is, the set of all points of X of norm one, while B(x, r) and B(x, r) denote, respectively, the open and closed balls centered at x with radius r. Some other notations are introduced as and when needed.

Homogeneous continuity
We begin with the following definition. Definition 2.1 Let X and Y be normed spaces and E ⊂ X. The mapping f : X → Y is said to be homogeneously continuous atx ∈ X on E if for every > 0 there exist δ > 0 and 0 < β ≤ 1 such that for all 0 < t ≤ β and all x, y ∈ E.
We are going to provide some sufficient conditions under which a mapping f is homogeneously continuous. Let us recall that a mapping f : X → Y is said to be locally Lipschitz aroundx ∈ X if there exist a neighborhood O ofx and a real number λ > 0 such that is uniformly continuous (E × (0, 1] equipped with the product topology with the usual linear operations of vector addition and scalar multiplication), then f is homogeneously continuous atx on E.
Proof Let > 0. By hypothesis, there exist δ, β > 0 such that for all for all x, y ∈ E with xy < δ and all 0 < t ≤ 1. Thus for all x, y ∈ E with xy < δ and all 0 < t ≤ 1. This completes the proof.

Generalized derivatives
We recall the definitions of the Hadamard and Gateaux derivatives: The Hadamard direc- where (ν n ) and (t n ) are any sequences such that ν n → ν and t n → 0 + .
The following facts are well known: • Hadamard differentiability is a stronger notion than Gateaux differentiability, see, e.g., [6]; when f is Hadamard differentiable atx, it is Gateaux (directional) differentiable at x and, moreover, f G (x) is continuous; • For locally Lipschitz mappings in normed spaces, Hadamard and Gateaux directional derivatives coincide. The following corollary uses Hadamard differentiability and provides another sufficient condition for a mapping f to be homogeneously continuous.
Since f is continuous and the Hadamard directional derivative of f atx in every direction is uniformly continuous. Now apply Proposition 2.1.
The following proposition illustrates our main motivation for introducing the homogeneously continuous mappings.
for all x, y ∈ E with xy < δ and all 0 < t ≤ β.
Proof The proof is obvious; we therefore omit it.
For a mapping f : X → Y, we consider the following notions of lower directional derivatives which are crucial to our approach: Note that we have for every ν ∈ X. We shall observe that if inf ν∈S f l (x)(ν) > 0 and f is homogeneously continuous atx on S, then f satisfies the property (1.2) above.

Main results
Throughout the remaining part of the discussion, unless specified otherwise, we assume that X is a finite-dimensional space and Y is an arbitrary normed space. We now are completely ready to state the main theorem of the paper. For a positive scalar α ∈ R, let S α := x ∈ X : x = α = αS.
for all x ∈ B(x, δ). In other words,x is a strongly regular point of f .

Corollary 2.2
Let f : X → Y be homogeneously continuous atx ∈ X on S. If there exists some κ > 0 such that inf ν∈S f 0 (x)(ν) > κ, then there exists δ > 0 such that  ⊂ B(0, r)). The condition Proof By hypothesis, f H (x)(ν) exists for every ν ∈ S. By continuity of f and · , it follows that for every ν ∈ S. It follows that inf ν∈S f l (x)(ν) > κ. Since X is finite dimensional, S is compact. Hence, f is homogeneously continuous atx ∈ X on S, by Corollary 2.1. Now apply Theorem 2.1.
The following example has been considered in [7] (Example 2.1). We shall prove that the origin is a regular point of the involved mapping f once again by Theorem 2.1.
We have S = {±1} and therefore f is homogeneously continuous at 0 on S. One may easily verify that It follows that inf s∈S f l (0)(s) = 2 π > 0. Hence, if 0 < κ < 2 π , then there exists δ > 0 such that for all |x| < δ, by Theorem 2.1. Hence, 0 is a strongly regular point of f . Since f is continuous, thus the subset f -1 (0) is closed and therefore the distance function dist(·, f -1 (0)) is Lipschitz around 0 (see [8, p. 11]). Hence, 0 is a regular point of f .
Hoffman's result is considered as the starting point of the theory of error bounds, theory that has been extended over the years to different contexts. We refer to [3,[11][12][13] and the references therein for the discussion of the fundamental role played by Hoffman bounds and more generally by error bounds in mathematical programming. As described, for example, in [14], they are used, for instance, in convergence properties of algorithms, in sensitivity analysis, in designing solution methods for nonconvex quadratic problems. When . . , k, are some given functionals and A : X → Y is a linear (continuous) mapping, we have the following result. Theorem 3.2 (Ioffe, 1979, [15]) There exists some κ > 0 such that for all x ∈ X.
. Then, Theorem 3.2 yields the following result.

Corollary 3.1
There exists some κ > 0 such that

4)
for all x ∈ X.
In this section, we apply Theorem 2.1 and establish similar estimates. We prove that there existsκ > 0 such that for all x ∈ X, where L : X → X is a linear mapping with Ker L = G. Our results also allow us to evaluate the constantκ. The details are as follows.
Proposition 3.1 Let A : X → Y be a linear mapping and x * i ∈ X * , i = 1, 2, . . . , k be given. Suppose that L : X → X is a linear mapping such that Ker L = G. Then where γ is a positive real number given by Hence, f is homogeneously continuous at [0] on S M , by Corollary 2.1. We also have The For all x ∈ X. Letting κ → γ in (3.8), we obtain the desired result.
Remark 3.1 The existence of the linear mapping L : X → X discussed in Proposition 3.1 is straightforward. Indeed, G is a closed subspace of X and X is separable, thus there exists a (continuous) linear mapping L : X → X with Ker L = G (see [17]). Of course, one can easily define L directly (without using [17]). To see this, suppose that dim X = n and let {e 1 , . . . , e j } be a linearly independent basis for the vector space G. By linear algebra, we can extend {e 1 , . . . , e j } to get a linearly independent basis for X (since G is a subspace of X). Let us denote this basis by {e 1 , . . . , e j , e j+1 , . . . , e n }. Now for every x := x 1 e 1 + · · · + x n e n ∈ X, define the mapping L : X → X as One can easily check that L is well-defined, linear, and Ker L = G. for all x ∈ X.
Proof Let X = Y, and x * i ≡ 0 for all 1 ≤ i ≤ k. Then, G = Ker A. Letting L := A in Proposition 3.1 yields the result.