\UseRawInputEncoding

The optimal partition for multiparametric semialgebraic optimization ^†^†thanks: Supported by the National Natural Science Foundation of China (11871118,12271061).

Zi-zong Yan Xiang-jun Li and Jin-hai Guo School of Information and Mathematics, Yangtze University, Jingzhou, Hubei, China([email protected]).School of Information and Mathematics, Yangtze University, Jingzhou, Hubei, China([email protected]).School of Information and Mathematics, Yangtze University, Jingzhou, Hubei, China([email protected]).

Abstract

In this paper we investigate the optimal partition approach for multiparametric conic linear optimization (mpCLO) problems in which the objective function depends linearly on vectors. We first establish more useful properties of the set-valued mappings early given by us [YLG20] for mpCLOs, including continuity, monotonicity and semialgebraic property. These properties characterize the notions of so-called linearity and nonlinearity subsets of a feasible set, which serve as stability regions of the partition of a conic (linear inequality) representable set. We then use the arguments from algebraic geometry to show that a semialgebraic conic representable set can be decomposed into a union of finite linearity and/or nonlinearity subsets. As an application, we investigate the boundary structure of the feasible set of generic semialgebraic mpCLOs and obtain several nice structural results in this direction, especially for the spectrahedon.

Keywords: Semialgebraic set, multiparametric conic linear optimization, conic representable set, set-valued mapping, optimal partition, nonlinearity set, transition face

AMS subject classifications. Primary: 90C31, 49J53; Secondary: 90C25, 14P10, 49K40

1 Introduction

This paper is mainly concerned with the optimal partition approach for multiparametric conic linear optimization (mpCLO) problems by means of the connections existing between the vectors of parameters. In particular, we focus on the sensitivity analysis of semialgebraic mpCLOs.

Formally we consider the following pair of mpCLOs

\begin{array}[]{ll}\min\limits_{x}&\langle c,x\rangle\\ s.t.&Ax=b,\\ &Mx=Md+v,\\ &x\in K\end{array}

(1)

and

\begin{array}[]{ll}\max\limits_{w,y}&b^{T}w\\ s.t.&A^{T}w+y=c+M^{T}u,\\ &w\in\mathbb{R}^{m},y\in K^{*},\end{array}

(2)

where $K\subset\mathbb{R}^{q}$ is a pointed, closed, convex, solid (with non-empty interior) cone (Ref. [NN94]), and $K^{*}$ is the dual of $K$ , that is,

K^{*}=\{y\in\mathbb{R}^{q}|\langle y,x\rangle\geq 0,\forall x\in K\}.

Hence $d,c\in\mathbb{R}^{q}$ , $b\in\mathbb{R}^{m}$ and $A\in\mathbb{R}^{m\times q}$ and $M\in\mathbb{R}^{r\times q}$ are fixed data, and $u,v\in\mathbb{R}^{r}$ are vectors of parameters. Such a pair of problems covers linear programming (LP) and semidefinite programming (SDP) problems. As an active research subject today, SDPs have gained a lot of attention from many practical applications in fields such as synthesis of filter and antennae arrays, structural design, stability analysis of mechanics, robust optimization, and relaxation of combinatorial optimization—topics of contemporary interest (Refs. [BN01, BV04, LV98, SY15, Tod01]).

A astonishing aspect is that the Lagrangian dual of the problem (2) is the same as a lifting of the problem (1), and they can be expressed in one unified form:

\begin{array}[]{ll}\min\limits_{x}&\langle c+M^{T}u,x\rangle\\ s.t.&Ax=b,\\ &x\in K,\end{array}

(3)

see [YLG20]. In the same way, both the Lagrangian dual of the problem (1) and a lifting of the problem (2) can be described as another unified form:

\begin{array}[]{ll}\max\limits_{w,s,y}&b^{T}w+(Md+v)^{T}s\\ s.t.&A^{T}w+M^{T}s+y=c,\\ &w\in\mathbb{R}^{m},s\in\mathbb{R}^{r},y\in K^{*}.\end{array}

(4)

Their optimality characterizations lead to a multiparametric Karush-Kuhn-Tucker (mpKKT) property. More precisely, a triple of vectors $(x;w,y)\in\mathbb{R}^{q\times r\times q}$ is optimal for the problems (1) and (2) if and only if it satisfies the following system of conic linear equations


		$\displaystyle Ax=b,\quad Mx=Md+v,\quad x\in K,$		(5a)
		$\displaystyle A^{T}w+y=c+M^{T}u,\quad w\in\mathbb{R}^{m},\quad y\in K^{*},$		(5b)
		$\displaystyle\langle x,y\rangle=0,$		(5c)

(see Remark LABEL:remkkt1 and Theorem LABEL:classical1 below), in which the equality (5c) is well known the complementary slackness property. The surprising aspect of this result is that it holds for some vector pairs $(u,v)\in\mathbb{R}^{r\times r}$ , under a Slater-type condition.

The mpKKT property (5a)-(5c) is of interest because it covers the classical KKT property of either the primal-dual problems (2) and (3) or (1) and (4) for some vector pairs $(u,v)=(-s,v)$ (see also Remark LABEL:remkkt1 below). This characterization result motivates us to explore in detail the relationship between the vectors of parameters $u$ and $v$ . This relationship can be read as follows.

Definition 1.1.

The set-valued mappings $\Phi,\Psi:\mathbb{R}^{r}\rightrightarrows\mathbb{R}^{r}$ are defined by


$\displaystyle\Phi(u)$	$\displaystyle=\{v\in\mathbb{R}^{r}\|\exists\ (\bar{x};\bar{w},\bar{y})\ \operatorname{s.t.\ the\ system}\ (\ref{kktcond1})-(\ref{kktcond3})\ \operatorname{holds}\},$	(6a)
$\displaystyle\Psi(v)$	$\displaystyle=\{u\in\mathbb{R}^{r}\|\exists\ (\bar{x};\bar{w},\bar{y})\ \operatorname{s.t.\ the\ system}\ (\ref{kktcond1})-(\ref{kktcond3})\ \operatorname{holds}\},$	(6b)

respectively.

We will show that under a Slater-type condition, the above definition are essentially equivalent to the one in the paper [YLG20] (see Subsection LABEL:oldnew1). The latter was used by us to discuss the behaviour of the optimal partition of a conic representable set.

Clearly, the domains of the set-valued mappings $\Psi$ and $\Phi$ are contained in the following two sets


$\displaystyle\varTheta_{P}$	$\displaystyle=\{v\in\mathbb{R}^{r}\|\exists\ \bar{x}\ \operatorname{s.t.\ the\ system\ (\ref{kktcond1})\ holds}\},$	(7a)
$\displaystyle\varTheta_{D}$	$\displaystyle=\{u\in\mathbb{R}^{r}\|\exists\ (\bar{w},\bar{y})\ \operatorname{s.t.\ the\ system\ (\ref{kktcond2})\ holds}\},$	(7b)

respectively, in which every element is naturally associated with the feasible sets of problems (1) and (2). One of the main reasons for considering them together is that these two sets enjoy more powerful results in the context of the optimal partition approach for mpCLOs (see Section LABEL:parti14).

Such sets can be represented as conic representable sets, including polyhedra and spectrahedra (see Lemma LABEL:conicr1 and the corresponding remark). Polyhedra enjoys considerable interest throughout pure and applied mathematics, and spectrahedra inherit many of the favorable properties of polyhedra. From the viewpoint of geometry, the boundary structure of spectrahedra is far more intricate than that of polyhedra. Recently there has been considerable interest in understanding the geometry of spectrahedra (Refs. [GW15, HV07, Hen11, Las17, NP10, Nie08, San15, Sch18]). Unfortunately, except for polyhedra, relatively little is known about the boundary structure of spectrahedra.

Like the classical KKT property, the mpKKT property (5a)-(5c) characterizes the optimality characteristics of the problems (1) and (2) whenever the complement slackness property holds (see Theorem LABEL:classical1 below). So in our terminology, the problems (1) and (2) are called a pair of primal-dual mpCLOs (actually, their lifting models (4) and (3) is another pair of primal-dual mpCLOs, see Figure 1). Correspondingly, the sets $\varTheta_{P}$ and $\varTheta_{D}$ are called a pair of primal-dual conic representable sets.

Refer to caption — Figure 1: Various different duals

The following is assumed to hold throughout this paper.

Assumption 1. Both the matrices $A$ and $M$ are of full rank, and the range spaces of their transpose don’t intersect, i.e., $R(A^{T})\cap R(M^{T})=\emptyset$ .

Assumption 2. There exists a (primal) feasible point $x\in\operatorname{int}(K)$ for the problem (3), and there exists a (dual) feasible point $(w,s,y)$ with $y\in\operatorname{int}(K^{*})$ for the problem (4).

Assumption 2 implies (see e.g. [YLG20]) that both $\varTheta_{P}$ and $\varTheta_{D}$ satisfy a Slater type condition, i.e., they have at least an interior point in $\mathbb{R}^{r}$ .

It should be noted that column vectors of $M^{T}$ denote perturbation directions of the objective function of (3). On the other hand, we [YLG20] assumed that $d$ and $(0,0,c)$ are feasible for the problems (3) and (4), respectively. Although this assumption is unnecessary for this paper, for convenience we still adopt the same notation of the paper [YLG20] to preserve the vector $d$ .

Motivated by our recent work [YLG20], the main subject of this paper is to make further study the optimal partition of the primal and dual conic representable sets defined by (7a)-(7b). For the optimal partition, an overview of the literature reveals that there are two major types of invariancy sets—linearity and nonlinearity sets—whose difference is whether the optimal objective value functions of the problems (1) and (2) are linear with respect to the vector of parameter $u$ or $v$ (see Theorems LABEL:linritys1 and LABEL:linritys2).

There are significant differences between polyhedra and specthedra although specthedra naturally generalize the class of polyhedra. Recall that an extreme point of a convex set is called a vertex if its normal cone is full-dimensional, and is called smooth if its normal cone is one-dimensional. For a polyhedron, extreme points and vertices coincide, and there are only finitely many of them. However, the unit ball $\{x\in\mathbb{R}^{q}|\|x\|\leq 1\}$ , which is affinely isomorphic to a specthedron, has uncountably many extreme points and no vertices when $q\geq 2$ (Ref. [MST16]). Linearity and nonlinearity sets facilitate the study of their boundary structure. For example, a linearity set is usually used to analyze shadow prices or marginal revenues for the sensitivity analysis of LPs (Refs. [GS55, Gal86]), whereas a nonlinearity set is in general regarded as a stability region and its identification has a great influence on the post-optimal analysis of SDPs (Refs. [CA17, CA18]).

For theoretical and practical significance the question of the distinction between linearity and nonlinearity sets in the description of a conic representable set is very important. Although the general question has so far been elusive, many results have been obtained in answer of the question for some special cases. For example, a linearity set is known to be convex for polyhedra [GG08], and a nonlinearity set is open for spectrahedra [YLG20]; a polyhedron has only linearity subsets, and a specthedron usually has a nonlinearity subset (see Corollary LABEL:ghui1 and the corresponding remarks). Such two types of invariancy sets were first distinguished by Mohammad-Nezhad and Terlaky [MT20] in the context of SDPs, although the first type has been studied extensively in LPs (Refs. [AD92, Deh07, Gre94, JR03]), quadratic programming and linear complementarity problems (Ref. [Ber97]).

In addition, multiparametric programming has emerged in the last two decades as an important optimization-based tool for systematically analyzing the effect of uncertainty and variability in optimization. There are plenty of works on multiparametric linear (Refs. [BK21, Gal95, GT72, Fil97, Sch87]), nonlinear (Refs.[Fia76, Fia83, FK86, Koj79]), quadratic (Refs. [BMP02, BD95, PD02, WW20]), mixed-integer linear programming (Refs. [DP00, GN77, JM06, MM75]) and others (Refs. [CJ91, MC80]). To the best of our knowledge, in the general case there are hardly any results regarding the question mentioned above; and even for the case of the two parameters ( $r=2$ ), the answer is unknown.

In this paper we exhibit more algebraic and geometric properties of the linearity and the nonlinearity sets than in the literature. They are based on supposedly good properties of the set-valued mappings $\Phi$ and $\Psi$ . We show that if $K$ is a semialgebraic set, then both $\Phi$ and $\Psi$ are semialgebraic mappings. That is, their graphs $gph(\Phi)$ and $gph(\Psi)$ on $\mathbb{R}^{r}\times\mathbb{R}^{r}$ are semialgebraic—those whose graphs can be written as a finite union of sets, each obtained by finitely many polynomial equalities and inequalities. We then use the arguments from algebraic geometry, in particular properties of semialgebraic sets, to show that a semialgebraic conic representable set can be decomposed into a union of finite linearity and/or nonlinearity subsets. Our results imply that the feasible set has finitely vertices, and in particular the spectrahedon usually has a smooth surface. These geometric properties have the practical significance in describing the boundary structure of the feasible set, especially for the spectrahedon. All these lead to a transparent and unified treatment of the boundary structure of the feasible sets of semialgebraic optimization problems, covering, in particular, polynomial optimization problems, LPs and SDPs.

The organization of this paper is as follows. In Section 2, we recall some preliminary results from set-valued analysis and semialgebraic geometry. In Section 3, we review the essential facts from [YLG20] about the set-valued mapping and discuss their continuity and monotonicity, and prove that they are semialgebraic mappings when $K$ is a semialgebraic set. In Section 4, we investigate the sensitivity of the optimal partition of semialgebraic conic representable sets and obtain several nice structural results in this direction. The main tool of our investigation comes from semialgebraic geometry. The last section contains some conclusions.

2 Preliminaries

2.1 Basic notation, terminology and facts

Throughout this paper we follow the terminology and notation of the book [SPR06, Roc70, RW09] and the paper [YLG20].

The symbol $\mathbb{R}^{q}$ denotes a $q$ -dimensional Eucildean space with the standard inner product $\langle\cdot,\cdot\rangle$ and the corresponding norm $\|x\|$ . A subset $C\subset\mathbb{R}^{q}$ is a convex if for any $x,y\in C$ and any $\lambda\in[0,1]$ , $\lambda x+(1-\lambda)y\in C$ , and a point $z\in C$ is an extreme point of $C$ if there is no any $\lambda\in(0,1)$ and $x,y\in C$ such that $z=\lambda x+(1-\lambda)y$ .

A subset $D\subset\mathbb{R}^{q}$ is a cone if it is closed under positive scalar multiplication. It is called pointed if $D$ contains no straight line, i.e., $x\in D,x\neq 0$ imply $-x\notin D$ .

For any subset $W\subset\mathbb{R}^{q}$ , denote $\operatorname{int}(W)$ , $\operatorname{cl}(W)$ and $\operatorname{conv}(W)$ as the interior, closure and convex hull of $W$ , respectively.

Let $C$ be a nonempty convex set in $\mathbb{R}^{q}$ . Given a boundary point $\bar{x}$ of $C$ , its normal cone $\operatorname{Normal}(C,\bar{x})$ is defined by

\operatorname{Normal}(C,\bar{x})=\{c\in\mathbb{R}^{q}|\langle c,\bar{x}\rangle\geq\langle c,x\rangle\ \operatorname{for\ all}\ x\in C\}.

For any $A\in\mathbb{R}^{m\times q}$ , the notation $\operatorname{rank}(A)$ and $R(A)$ denote the rank and the range space of the matrix $A$ . In particular, if $m=q$ and $A^{2}=A=A^{T}$ , then $A$ is called a symmetric projection matrix.

Note that the feasible sets of the problems (3) and (4), denoted by $\mathscr{X}$ and $\mathscr{Y}$ , respectively, do not depend on the given vectors of parameters $u$ and $v$ . Without causing confusion, the optimal solutions are denoted by $x^{*}(u)$ and $(w^{*}(v),s^{*}(v),y^{*}(v))$ , respectively, if they exist. That is,

	$\displaystyle x^{*}(u)$	$\displaystyle\in$	$\displaystyle\mathscr{X}^{*}(u)=\arg\min\{\langle c+M^{T}u,x\rangle\|x\in\mathscr{X}\},$
	$\displaystyle(w^{}(v),s^{}(v),y^{*}(v))$	$\displaystyle\in$	$\displaystyle\mathscr{Y}^{*}(v)=\arg\max\{b^{T}w+(Md+v)^{T}s\|(w,s,y)\in\mathscr{Y}\}.$

For brevity, sometimes, we also replace $(w^{*}(v),s^{*}(v),y^{*}(v))\in\mathscr{Y}^{*}(v)$ by $y^{*}(v)\in\mathscr{Y}^{*}(v)$ .