Improving the Approximated Projected Perspective Reformulation by dual information

Abstract

We propose an improvement of the Approximated Projected Perspective Reformulation (AP${}^2$R) for dealing with constraints linking the binary variables. The new approach solves the Perspective Reformulation (PR) once, and then use the corresponding dual information to reformulate the problem prior to applying AP${}^2$R, thereby combining the root bound quality of the PR with the reduced relaxation computing time of AP${}^2$R. Computational results for the cardinality-constrained Mean–Variance portfolio optimization problem show that the new approach is competitive with state-of-the-art ones.

Introduction

We study solution techniques for convex separable Mixed-Integer Non-Linear Programs (MINLP) with $n$ semi-continuous variables $x_i\in \mathbb{R}$ for $i\in N=\{1,\dots ,n\}$ which either assume the value $0$ or lie in the interval ${\mathcal{X}}_i=[{\underline{x}}_i,{\bar{x}}_i]$ ($-\infty <{\underline{x}}_i<{\bar{x}}_i<\infty$). This can be expressed, introducing $y_i\in \{0,1\}$ for $i\in N$, as

$$\text{(P)}minh\left(z\right)+\sum _{i\in N}{f}_i\left(x_i\right)+c_i{y}_i$$$$Ax+By+Cz=b$$$$(x,z)\in \mathcal{O}$$$${\underline{x}}_i{y}_i\le x_i\le {\bar{x}}_i{y}_i,y_i\in {[0,1]}^n,x_i\in {\mathbb{R}}^{n}i\in N$$$$y_i\in \mathbb{Z}i\in N.$$

We assume the functions $f_i$ to be closed convex, one time continuously differentiable and finite in the interval $({\underline{x}}_i,{\bar{x}}_i)$; w.l.o.g. we also assume $f_i\left(0\right)=0$. In (P) we single out the linking constraints (2) that contain all the relationships linking the $y_i$ variables among them and with the other variables of the problem, except those (4) that “define” the semi-continuous nature of the $x_i$. The reformulation technique developed in [7] require (2) to be empty; the extension developed in [1] allows to overcome this limitation, but potentially at the cost of a worse bound quality. The aim of this paper is to deal with constraint (2) in a cost-effective way. For our approach to work, (2) must have a compatible structure with that of (1); we initially assume equality constraints, with extensions discussed in Section 3. Because our approach hinges on availability of dual information for the continuous relaxation, we assume that the function $h(\cdot )$ in the “other variables $z$” and the “other constraints (3)” are convex, i.e., (P) is a convex MINLP. Actually, in many applications everything but (1) is linear. It will be sometimes expedient to refer to (3)–(4) as “$(x,y,z)\in \mathcal{P}$”, and to $\underline{\mathcal{P}}$ as the set obtained by $\mathcal{P}$ relaxing integrality constraints on $z$ and $x$, if any.

Often, the most pressing issue in solving (P) is to derive tight lower bounds on its optimal value $\nu$(P), which is typically done by solving its (convex) continuous relaxation (P) (we denote by $\nu$(X) and (X), respectively, the optimal value and the continuous relaxation of any problem (X)). However, often $\nu$(P) $\ll \nu$(P), making the solution approaches inefficient. The presence of semi-continuous variables has been exploited to propose reformulations (P${}^{\prime }$) of (P) such that $\nu$(P) $=\nu$(P${}^{\prime }$) $\ge \nu$(P ${}^{\prime }$) $\gg \nu$(P). This starts from considering (1) as $h\left(z\right)+{\sum }_{i\in N}{f}_i(x_i,y_i)$, where $f_i(x_i,y_i)=f_i\left(x_i\right)+c_i$ if $y_i=1$ and ${\underline{x}}_i\le x_i\le {\bar{x}}_i$, $f_i(0,0)=0$, and $f_i(x_i,y_i)=\infty$ otherwise. The convex envelope of $f_i(x_i,y_i)$ is known [4] to be ${\tilde{f}}_i(x_i,y_i)=y_i{f}_i(x_i∕y_i)+c_i{y}_i$ – using the perspective function of $f_i$ – which yields the Perspective Reformulation of (P)

$$\text{(PR)}min\left\{h\left(z\right)+\sum _{i\in N}{\tilde{f}}_i(x_i,y_i):\left(2\right),(x,y,z)\in \mathcal{P},\left(5\right)\right\}.$$

As $f_i$ is convex, ${\tilde{f}}_i$ is convex for $y_i\ge 0$; since $x_i=0$ if $y_i=0$, ${\tilde{f}}_i$ can be extended by continuity assuming $0f_i(0∕0)=0$. Hence, (PR) is a convex MINLP if (P) is. Its continuous relaxation (PR)—the Perspective Relaxation of (P)—usually has $\nu$(PR) $\gg \nu$(P), making (PR) a more convenient formulation [8], [9]. If $f_i$ is SOCP-representable then so is ${\tilde{f}}_i$, hence the PR of a Mixed-Integer Second-Order Cone Program (MI-SOCP) is still a MI-SOCP. Thus, (PR) is not necessarily more complex to solve – and, sometimes, even less so [2]– than (P). Alternatively, one can consider a Semi-Infinite MINLP reformulation of (PR) where Perspective Cuts [4]– linear outer approximations of the epigraph of ${\tilde{f}}_i$ – are dynamically added. This is often the best approach [6], in particular for “general” (P) where no other structure is available. It is appropriate to remark that the (PR) approach also applies if the $x_i$ are vectors such that $y_i=0⟹x_i=0$ and $y_i=1⟹x_i\in {\mathcal{X}}_i$, with ${\mathcal{X}}_i$ a polytope; yet, here, as in [1], [7], each $x_i$ must be a single variable.

While (PR) provides a better bound, it is also usually more time consuming to solve than (P) because ${\tilde{f}}_i$ is “more complex” than $f_i$. This trade-off is nontrivial, in particular if $f_i$ is “simple”. For instance, if $f_i$ is quadratic and everything else is linear, (P) is a Mixed-Integer Quadratic Program (MIQP) whereas (PR) is a MI-SOCP; hence, (P) – a QP – can be significantly cheaper to solve than (PR)—a SOCP. The Projected PR (P${}^2$R) idea underpinning the approach studied here was indeed proposed in [7] for the quadratic case, and ${\underline{x}}_i\ge 0$. It was then extended in [1] to a more general class of functions, and allowing ${\underline{x}}_i<0$. However, ${\underline{x}}_i<0<{\bar{x}}_i$ renders some of the arguments significantly more complex, hence for the sake of simplicity we will only present here the case where ${\underline{x}}_i\ge 0$; it will be plain to see that the arguments immediately extend to the more general one. The P${}^2$R idea is to analyze ${\tilde{f}}_i$ as a function of $x_i$ only, i.e., projecting away $y_i$: under appropriate assumptions, and if there are no linking constraints (2), this turns out to be a piecewise-convex functions with a “small” number of pieces, that can be characterized by just looking at the data of (P) (cf. (7)). Hence, (PR) can be reformulated in terms of piecewise-convex objective functions, which makes it easier to solve, especially when $\mathcal{O}$ has some valuable structure (e.g., flow or knapsack) [7]. However, in several applications (2) are indeed present [1], [3], [4], [8], [10]. Furthermore, since the binary variables $y_i$ are removed from the formulation, branching has to be done “indirectly”, which rules out using off-the-shelf solvers. To overcome these two limitations, in [1] the Approximated P${}^2$R (AP${}^2$R) reformulation has been proposed whereby the $y_i$, after having been eliminated, are re-introduced in the formulation in order to encode the piecewise nature of ${\tilde{f}}_i$. This is possible even if (2) are present, and it has the advantage that (AP${}^2$R) is still a MIQP if (P) is. However, $\nu$(AP²R) $<\nu$(PR) may, and does, happen when linking constraints (2) are present, whence the “Approximate” moniker. This is still advantageous in some cases, but it may happen that the weaker bounds outweigh the faster solution time, making the approach not competitive with more straightforward implementations of the PR [1].

The aim of this paper is to improve the AP${}^2$R by presenting a simple and effective way to ensure that $\nu$(AP²R) $=\nu$(PR) even if (2) are present, while keeping the shape of the formulation – and therefore, hopefully, the cost of (AP²R) – exactly the same. Since bound equivalence only holds at the root node of the B&C it is not obvious that the approach, despite the quicker solution times of (AP²R), is competitive. However, this is shown to be true in at least one relevant application, the Mean–Variance problem (with min buy-in and cardinality constraints) in portfolio optimization.