Characterizations of robust and stable duality for linearly perturbed uncertain optimization problems

TL;DR

This paper develops a robust optimization framework with duality relations that account for uncertainty, providing characterizations of zero-duality gap using epsilon-minima and epsilon-subdifferentials, applicable to conic and infinite optimization.

Contribution

It introduces a novel robust duality model for uncertain optimization problems, linking duality properties to epsilon-minima and epsilon-subdifferentials, and analyzes extreme cases.

Findings

Characterizes robust zero-duality gap using epsilon-minima.

Provides duality relations for uncertain conic and infinite optimization.

Analyzes classical and extreme duality cases in detail.

Abstract

We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail.

Equations

$$F_u:X\times Y_u\to {\mathbb{R}}_{\infty }:=\mathbb{R}\cup \{+\infty \},\text{ with }u\in U,$$

$$({\text{P}}_u){}_{x^{\ast }}:\underset{x\in X}{\inf }\left\{F_u(x,0_u)-⟨x^{\ast },x⟩\right\}.$$

$${(\mathrm{RP})}_{x^{\ast }}:\underset{x\in X}{\inf }\left\{\underset{u\in U}{\sup }{F}_u(x,0_u)-⟨x^{\ast },x⟩\right\}.$$

$$F_{u}^{\ast}(x^{\ast},y_{u}^{\ast}):=\sup\limits_{(x,y_{u})\in X\times Y_{u}}\Big{\{}\langle x^{\ast},x\rangle+\langle y_{u}^{\ast},y_{u}\rangle-F_{u}(x,y_{u})\Big{\}},\quad(x^{\ast},y_{u}^{\ast})\in X^{\ast}\times Y_{u}^{\ast},$$

$$({\text{D}}_u){}_{x^{\ast }}:\underset{y_u^{\ast }\in Y_u^{\ast }}{\sup }-F_u^{\ast }(x^{\ast },y_u^{\ast }).$$

$$\sup ({\text{D}}_u{)}_{x^{\ast }}\le \inf ({\text{P}}_u{)}_{x^{\ast }}\le \inf (\text{RP}{)}_{x^{\ast }},\forall u\in U.$$

$$(\text{ODP}{)}_{x^{\ast }}:\underset{(u,y_u^{\ast })\in \Delta }{\sup }-F_u^{\ast }(x^{\ast },y_u^{\ast }),$$

$$\sup (\mathrm{ODP}{)}_{x^{\ast }}=\underset{u\in U}{\sup }{\left({\mathrm{D}}_u\right)}_{x^{\ast }}\le \inf {(\mathrm{RP})}_{x^{\ast }}.$$

$$p:=\underset{u\in U}{\sup }{F}_u(\cdot ,0_u).$$

$$\underset{x\in X}{\inf }\underset{u\in U}{\sup }\left\{F_u\left(x,0_u\right)-⟨x^{\ast },x⟩\right\}=\underset{\left(u,y_{{}^u}^{\ast }\right)\in \Delta }{\sup }-F_u^{\ast }\left(x^{\ast },y_u^{\ast }\right).$$

$${(\mathrm{P})}_{x^{\ast }}:\underset{x\in X}{\inf }\left\{f(x)-⟨x^{\ast },x⟩\right\},$$

$$\underset{x\in X}{\inf }\left\{F\left(x,0_Y\right)-⟨x^{\ast },x⟩\right\}=\underset{y^{\ast }\in Y^{\ast }}{\sup }-F^{\ast }\left(x^{\ast },y^{\ast }\right),$$

$$(\mathrm{P}):{\left\{\underset{x\in X}{\inf }{f}_u(x)\right\}}_{u\in U},$$

$$\text{(RP)}\underset{x\in X}{\inf }\underset{u\in U}{\sup }{f}_u(x).$$

$$F_{u}^{\ast}\left({x,y_{u}^{\ast}}\right)=\left\{\begin{array}[]{ll}{f_{u}^{\ast}\left({x^{\ast}}\right),}&\mathrm{if}\;\;{y_{u}^{\ast}}=0_{u}^{\ast},\\ {+\infty,}&\mathrm{if}\;\;y_{u}^{\ast}\neq 0_{u}^{\ast}.\end{array}\right.$$

$${\left(\underset{u\in U}{\sup }{f}_u\right)}^{\ast }(x^{\ast })=\underset{u\in U}{\inf }{f}_u^{\ast }(x^{\ast }),$$

$$\underset{x\in X}{\inf }\underset{u\in U}{\sup }{f}_u(x)=\underset{u\in U}{\sup }\underset{x\in X}{\inf }{f}_u(x).$$

$$(\mathrm{P}):{\left\{\underset{x\in X}{\inf }f(x)\text{s.t.}{H}_u(x)\in -S_u\right\}}_{u\in U},$$

$$(\mathrm{P}):{\left\{\underset{x\in {\mathbb{R}}^n}{\inf }f(x)=-c(x)+{\mathrm{i}}_{{\mathbb{R}}_{+}^n}(x)\text{s.t.}{A}_{u}x-b_u\in -{\mathbb{R}}_{+}^{m_u}\right\}}_{u\in U},$$

$${F_{u}}(x,{y_{u}})=\left\{{\begin{array}[]{ll}f{(x),}&{\mathrm{if}\;{H_{u}}(x)+{y_{u}}\in-{S_{u}}},\\ {+\infty,}&{\mathrm{else}}.\end{array}}\right.$$

$$(\mathrm{RP}):\underset{x\in X}{\inf }f(x)\text{s.t.}{H}_u(x)\in -S_u,\forall u\in U.$$

$$F_{u}^{\ast}({x^{\ast}},{y}_{u}^{\ast})=\left\{{\begin{array}[]{ll}{{{\left({f+y_{u}^{\ast}\circ{H_{u}}}\right)}^{\ast}}({x^{\ast}}),}&{\mathrm{if}\;y_{u}^{\ast}\in S_{u}^{+},}\\ {\ +\infty,}&{\mathrm{else,}}\end{array}}\right.$$

$$(\mathrm{ODP}):\underset{u\in U,y_u^{\ast }\in S_u^{+}}{\sup }\underset{x\in X}{\inf }\left\{f(x)+⟨y_u^{\ast },H_u(x)⟩\right\}$$

$$\inf \left\{f(x):H_u(x)\in -S_u,\forall u\in U\right\}=\underset{\genfrac{}{}{0ex}{}{u\in U\text{hfill}}{y_u^{\ast }\in S_u^{+}\text{hfill}}}{\max }\underset{x\in X}{\inf }\left\{f(x)+⟨y_u^{\ast },H_u(x)⟩\right\}.$$

$$(\mathrm{P}):{\left\{\underset{x\in X}{\inf }f(x)s.t.g_t(x)\le 0,\forall t\in S\right\}}_{S\in U}.$$

$${F_{S}}\left({x,{\mu}}\right)=\left\{{\begin{array}[]{ll}{f(x),}&{\mathrm{if}\;{g_{s}}(x)\leq-{\mu_{s}},\;\forall s\in S,}\\ {+\infty,}&{\mathrm{else}}.\end{array}}\right.$$

$$F_{S}^{\ast}\left({{x^{\ast}},{\lambda}}\right)=\left\{{\begin{array}[]{ll}{{\left({f+\sum\limits_{s\in S}{{\lambda_{s}g_{s}}}}\right)^{\ast}}({x^{\ast}}),}&{\mathrm{if}\;{\lambda}\in\mathbb{R}_{+}^{S},}\\ {+\infty,}&{\mathrm{else}}.\end{array}}\right.$$

$${(\mathrm{RP})}_{0_X^{\ast }}:\inf f(x)s.t.g_t(x)\le 0\text{ for all }t\in T,$$

$${(\mathrm{ODP})}_{0_X^{\ast }}:\underset{S\in U,\lambda \in {\mathbb{R}}_{+}^S}{\sup }\left\{\underset{x\in X}{\inf }\left(f(x)+\sum _{s\in S}{\lambda }_s{g}_s(x)\right)\right\},$$

$${(\mathrm{ODP})}_{0_{X^{\ast }}}:\underset{\lambda \in {\mathbb{R}}_{+}^{(T)}}{\sup }\left\{\underset{x\in X}{\inf }\left(f(x)+\sum _{t\in T}{\lambda }_t{g}_t(x)\right)\right\},$$

Full text

Characterizations of robust and stable duality for linearly

perturbed uncertain optimization problems

N. Dinh,   M.A. Goberna,   M.A. Lopez,   M. Volle

International University, Vietnam National University - HCMC, Linh Trung ward, Thu Duc district, Ho Chi Minh city, Vietnam (ndinh@hcmiu.edu.vn). Department of Mathematics, University of Alicante, Spain (mgoberna@ua.es)Department of Mathematics, University of Alicante, Spain, and CIAO, Federation University, Ballarat, Australia (marco.antonio@ua.es) Avignon University, LMA EA 2151, Avignon, France (michel.volle@univ-avignon.fr)

Abstract. We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail.

Key words. Robust duality, strong robust duality, reverse strong robust duality, min-max robust duality.

2000 Mathematics Subject Classification. 90C26, 90C46, 90C31

1 Introduction

In this paper we consider a family of perturbation functions

[TABLE]

and where $X$ and $Y_{u},$ $u\in U,$ are given locally convex Hausdorff topological vector spaces (briefly, lcHtvs), the index set $U$ is called the uncertainty set of the family, $X$ is its decision space, and each $Y_{u}$ is a parameter space. Note that our model includes a parameter space $Y_{u},$ depending on $u\in U,$ which is a novelty with respect to the “classical” robust duality scheme (see [17] and references therein, where a unique parameter space $Y$ is considered), allowing us to cover a wider range of applications including uncertain optimization problems under linear perturbations of the objective function. The significance of our approach is illustrated along the paper by relevant cases extracted from deterministic optimization with linear perturbations, uncertain optimization without linear perturbations, uncertain conic optimization and infinite optimization.** **The antecedents of the paper are described in the paragraphs devoted to the first two cases in Section 2.

We associate with each family $\left\{F_{u}:u\in U\right\}$ of perturbation functions corresponding optimization problems whose definitions involve continuous linear functionals on the decision and the parameter spaces. We denote by $0_{X},$ ${0}_{{}_{X}}^{\ast},$ $0_{u},$ and $0_{u}^{\ast},$* the null vectors of $X,$ its topological dual $X^{\ast},$ $Y_{u},$ and its topological dual $Y_{u}^{\ast},$ respectively. The optimal value of a minimization (maximization, respectively) problem $\mathrm{(}$P$\mathrm{)}$ is denoted by $\inf\mathrm{(}$P$\mathrm{)}$ ($\sup\mathrm{(}$P$\mathrm{)}$); in particular, we write $\min\mathrm{(}$P$\mathrm{)}$ ($\max\mathrm{(}$P$\mathrm{)}$) whenever the optimal value of $\mathrm{(}$P$\mathrm{)}$ is attained. We adopt the usual convention that $\inf\mathrm{(P)=+\infty}$ ($\sup\mathrm{(}$P$\mathrm{)=-\infty}$)* when the problem $\mathrm{(P)}$** has no feasible solution.** **The associated optimization problems are the following:

• •

Linearly perturbed uncertain problems: for each $(u,x^{\ast})\in U\times X^{\ast},$

[TABLE]

• •

Robust counterpart of $\left\{\mathrm{(}P_{u}\mathrm{)}{{}_{{x^{\ast}}}}\right\}_{u\in U}:$

[TABLE]

Denoting by $F_{u}^{\ast}:X^{\ast}\times Y_{u}^{\ast}\rightarrow\overline{\mathbb{R}}$, where $\overline{\mathbb{R}}:=\mathbb{R\cup\{\pm\infty\}}$, the Fenchel conjugate of $F_{u}$, namely,

[TABLE]

we now introduce the corresponding dual problems:

• •

Perturbational dual of ${(\text{{P}}_{u})_{{x^{\ast}}}}:$

[TABLE]

Obviously,

[TABLE]

• •

Optimistic dual of $\mathrm{(}$RP$\mathrm{)}_{x^{\ast}}:$

[TABLE]

where $\Delta:=\left\{{\left({u,y_{u}^{\ast}}\right):u\in U,\ {y^{\ast}}\in Y_{u}^{\ast}}\right\}$ is the disjoint union of the spaces ${Y_{u}^{\ast}}$. We have

[TABLE]

We are interested in the following desirable robust duality properties:

$\bullet$ Robust duality is said to hold at $x^{\ast}$ if $\inf{\mathrm{(RP)}_{{x^{\ast}}}}=\sup{(\mathrm{ODP})_{{x^{\ast}}}}$,

$\bullet$ Strong robust duality at $x^{\ast}$ means $\inf{\mathrm{(RP)}_{{x^{\ast}}}}=\max{(\mathrm{ODP})_{{x^{\ast}}}}$,

$\bullet$ *Reverse strong robust duality *at $x^{\ast}$ means $\min{\mathrm{(RP)}_{{x^{\ast}}}}=\sup{(\mathrm{ODP})_{{x^{\ast}}}}$,

$\bullet$ Min-max robust duality at $x^{\ast}$ means $\min{\mathrm{(RP)}_{{x^{\ast}}}}=\max{(\mathrm{ODP})_{{x^{\ast}}}}$.

Each of the above desirable properties is said to be stable when it holds for any $x^{\ast}\in X^{\ast}$. The main results of this paper characterize these properties in terms of formulas involving the $\varepsilon$-minimizers and $\varepsilon$-subdifferentials of the objective function of the robust counterpart problem (RP)${}_{0_{X}^{\ast}}$, namely, the function

[TABLE]

Theorem 3.1 characterizes robust duality at a given point $x^{\ast}\in X^{\ast}$ as a formula for the inverse mapping of the $\varepsilon$-subdifferential at $x^{\ast}$ without any convexity assumption. The same is done in Theorem 4.1 to characterize strong robust duality. In the case when a primal optimal solution does exist we give a formula for the exact minimizers of $p-x^{\ast}$ to characterize dual strong (resp. min-max) robust duality at $x^{\ast}$, see Theorem 5.1 (resp. Theorem 5.2). We show that stable robust duality gives rise** **to a formula for the $\varepsilon$-subdifferential of $p$ (Theorem 6.1, see also Theorem 3.1). The same is done for stable strong robust duality (Theorem 7.1). A formula for the exact subdifferential of $p$ is provided in relation with robust duality at appropriate points (Theorem 8.1). The most simple possible formula for the exact subdifferential of $p$ (the so-called Basic Robust Qualification condition) is studied in detail in Theorem 8.2. All the results from Sections 1-8 are specified for the two extreme cases (the case with no uncertainty and the one in absence of linear perturbations), namely, Cases 1 and 2 in Section 2 (for the sake of brevity, we do not give the specifications for Cases 3 and 4). It is worth noticing the generality of the mentioned results (as they do not require any assumption on the involved functions) and the absolute self containment of their proofs. The use of convexity in the data will be addressed in a forthcoming paper.

2 Special Cases and Applications

In this section we make explicit the meaning of the robust duality of the general model introduced in Section 1, composed by a family of perturbation functions together with its corresponding optimization problems. We are doing this by exploring the extreme case with no uncertainty, the extreme case in absence of perturbations, and two other significant situations. In all these cases, we propose ad hoc families of perturbation functions allowing to apply the duality results to given optimization problems, either turning back to variants of well-known formulas for conjugate functions or proposing** **new ones.

Let us recall the robust duality formula, $\inf{\mathrm{(RP)}_{{x^{\ast}}}=}\sup\mathrm{(}\mathrm{ODP}\mathrm{)}_{x^{\ast}},$ i.e.,

[TABLE]

We firstly study the two extreme cases: the case with no uncertainty and the one with no perturbations.

Case 1. The case with no uncertainty: Deterministic optimization with linear perturbations deals with parametric problems of the form:

[TABLE]

where $f:X\rightarrow\mathbb{R}_{\infty}$ (i.e., $f\in(\mathbb{R}_{\infty})^{X}$) is the nominal objective function and the parameter is ${x^{\ast}\in X}^{\ast}.$ Taking a singleton uncertainty set $U=\left\{u_{0}\right\},$ $Y_{u_{0}}=Y$ and F_{u_{0}}=F\such that ${F\left({x,{0_{Y}}}\right)={f}(x)}$ for all $x\in X,$ (2.1) reads

[TABLE]

which is the fundamental perturbational duality formula [3], [20], [24]. Stable and strong robust duality theorems are given in [5] (see also [7] and [16] for infinite optimization problems).

Case 2. The case with no linear perturbations: Uncertain optimization without linear perturbations deals with families of problems of the form

[TABLE]

where $f_{u}\in(\mathbb{R}_{\infty})^{X},$ $u\in U.$ Taking F\such that $F_{u}(x,0_{u})=f_{u}(x)$ for all $u\in U,$ the problem $\mathrm{(}$P${}_{u}\mathrm{)}{{}_{{0}_{{}_{X}}^{\ast}}}$ represents here the scenario of $\mathrm{(P)}$ corresponding to $u\in U,$ while ${\mathrm{(RP)}_{{0}_{{}_{X}}^{\ast}}}$ is the robust counterpart (also called pessimistic or minmax counterpart in the robust optimization literature) of $\mathrm{(P),}$ namely,

[TABLE]

For instance, if F_{u}(x,y_{u})=f_{u}(x),\for all $y_{u}\in Y_{u},$ and $\operatorname{dom}f_{u}\neq\emptyset$, we have

[TABLE]

Then (2.1) writes

[TABLE]

which amounts, for $x^{\ast}={0}_{{}_{X}}^{\ast},$ to the $\inf-\sup$* duality in robust optimization,* also called robust infimum (recall that any constrained optimization problem can be reduced to an unconstrained one by summing up the indicator function of the feasible set to the objective function):

[TABLE]

Robust duality theorems without linear perturbations are given in [23] for a special class of uncertain non-convex optimization problems while [7] provides robust strong duality theorems for uncertain convex optimization problems which are expressed in terms of the closedness of suitable sets regarding the vertical axis of $X^{\ast}\times\mathbb{R}.$

Case 3. Conic optimization problem with uncertain constraints: Consider the uncertain problem

[TABLE]

where, for each $u\in U$, $S_{u}$ is an ordering convex cone in ${Y}_{u},$ $H_{u}\colon X\rightarrow{Y}_{u}$, and $f\in(\mathbb{R}_{\infty})^{X}$. Denote by ${S_{u}^{+}:=}\left\{{y_{u}^{\ast}\in Y}_{u}^{\ast}:\left\langle{y_{u}^{\ast},y_{u}}\right\rangle\geq 0,\forall{y_{u}\in}S_{u}\right\}$ the dual cone of $S_{u}$.

Problems of this type arise, for instance, in the production planning of firms producing $n$ commodities from uncertain amounts of resources by means of technologies which depend on the available resources (e.g., the technology differs when the energy is supplied by either fuel gas or a liquid fuel). The problem associated with each parameter $u\in U$ consists of maximizing the cash-flow $c\left(x_{1},...,x_{n}\right)$ of the total production, with $x_{i}$ denoting the production level of the $i$-th commodity, $i=1,..,n.$ The decision vector $x=\left(x_{1},...,x_{n}\right)$ must satisfy a linear inequality system $A_{u}x\leq b_{u},$ where the matrix of technical coefficients $A_{u}$ is $m_{u}\times n$ and $b_{u}\in\mathbb{R}^{m_{u}},$ for some m_{u}\in\mathbb{N}.\Denoting by $\mathrm{i}_{\mathbb{R}_{+}^{n}}$ the indicator function of $\mathbb{R}_{+}^{n}$ (i.e., $\mathrm{i}_{\mathbb{R}_{+}^{n}}(x)=0,$ when $x\in\mathbb{R}_{+}^{n},$ and $\mathrm{i}_{\mathbb{R}_{+}^{n}}(x)=+\infty,$ otherwise), the uncertain production planning problem can be formulated as

[TABLE]

with the space $Y_{u}=\mathbb{R}^{m_{u}}$ depending on the uncertain parameter $u.$

For each $u\in U$, define the perturbation function

[TABLE]

On the one hand, $\mathrm{(RP)}_{0_{X}^{\ast}}$ collapses to the robust counterpart of $\mathrm{(P)}$ in the sense of robust conic optimization with uncertain constraints:

[TABLE]

On the other hand, it is easy to check that

[TABLE]

$\mathrm{(ODP)}_{0_{X}^{\ast}}$ is nothing else than the optimistic dual in the sense of uncertain conic optimization:

[TABLE]

(a special case when $Y_{u}=Y$, $S_{u}=S$ for all $u\in U$ is studied in [8, page 1097] and [17]). Thus,

$\bullet$ Robust duality holds at $0_{X}^{\ast}$ means that $\inf\mathrm{(RP)}=\sup\mathrm{(}\mathrm{ODP}),$

$\bullet$ Strong robust duality holds at $0_{X}^{\ast}$ means that

[TABLE]

Conditions for having such an equality are provided in [8, Theorem 6.3], [9, Corollaries 5, 6], for the particular case $Y_{u}=Y$ for all $u\in U$.

Strong robust duality and uncertain Farkas lemma: We focus again on the case where $Y_{u}=Y$ and $S_{u}=S$ for all $u\in U$. For a given $r\in\mathbb{R}$, let us** **consider the following statements:

• (i)

$H_{u}(x)\in-S,\;\forall u\in U\quad\Longrightarrow\quad f(x)\geq r$,

• (ii)

$\exists u\in U,\exists{y_{u}^{\ast}}\in S^{+}$ such that $f(x)+\left\langle{y_{u}^{\ast}},H_{u}(x)\right\rangle\geq r,\;\forall x\in X.$

Then, it is true that the Strong robust duality holds at $0_{X}^{\ast}$ if and only if $\left[(i)\Longleftrightarrow(ii)\right]$ for each $r\in\mathbb{R},$ which can be seen as an uncertain Farkas lemma. For details** see [8, Theorem 3.2] (also [9, Corollary 5 and Theorem 1]). **

It is worth noticing that when return to problem $\mathrm{(P)}$, a given robust feasible solution $\overline{x}$ is a minimizer if and only if $f(\overline{x})\leq f(x)$ for any robust feasible solution $x.$ So, a robust (uncertain) Farkas lemma (with $r=f(\bar{x})$) will lead automatically to an optimality test for $\mathrm{(P).}$ Robust conic optimization problems are studied in [2] and [21].

Case 4. Discretizing infinite optimization problems: Let $f\in(\mathbb{R}_{\infty})^{X}$ and $g_{t}\in\mathbb{R}^{X}\;$for all$\;t\in T$ (a possibly infinite index set). Consider the set $U$ of nonempty finite subsets of $T,$ interpreted as admissible perturbations of $T,$ and the parametric optimization problem

[TABLE]

Consider the parameter space $Y_{s}:={\mathbb{R}^{S}}$ (depending on $S$) and the perturbation function ${F_{S}}:X\times{\mathbb{R}^{S}}\rightarrow\mathbb{R}_{\infty}$ such that, for any $x\in X$ and ${{{{\mu:=}({\mu_{s}})}_{s\in S}\in}\mathbb{R}^{S},}$

[TABLE]

We now interpret the problems associated with the family of function perturbations $\left\{{F_{S}:S\in U}\right\}.$ One has $Y_{s}^{\ast}={\mathbb{R}^{S}}$ and

[TABLE]

The robust counterpart at $0_{X}^{\ast},$

[TABLE]

is a general infinite optimization problem while the optimistic dual at $0_{X}^{\ast}$ is

[TABLE]

or, equivalently, the Lagrange dual of $\mathrm{(RP)}_{0_{X}^{\ast}},$ i.e.,

[TABLE]

where, for each $\lambda=\left({\lambda_{t}}\right)_{t\in T}\in\mathbb{R}_{+}^{(T)}$ (the subspace of $\mathbb{R}^{T}$ formed by the functions $\lambda$ whose support, supp$\lambda:=\left\{t\in T:{{\lambda_{t}\neq 0}}\right\},$ is finite),

[TABLE]

Following [10, Section 8.3], we say that $\mathrm{(RP)}_{0_{X}^{\ast}}$ is discretizable if there exists a sequence $\left(S_{r}\right)_{r\in\mathbb{N}}\subset U$ such that

[TABLE]

and it is *reducible *if there exists $S\in U$ such that

[TABLE]

Obviously, $\inf\mathrm{(RP)}_{0_{X}^{\ast}}=-\infty$ entails that $\mathrm{(RP)}_{0_{X}^{\ast}}$ is reducible which, in turn, implies that $\mathrm{(RP)}_{0_{X}^{\ast}}$ is discretizable.

Discretizable and reducible problems are important in practice. Indeed, on the one hand, discretization methods generate sequences $\left(S_{r}\right)_{r\in\mathbb{N}}\subset U$ satisfying 2.5 when $\mathrm{(RP)}_{0_{X}^{\ast}}$ is discretizable; discretization methods for linear and nonlinear semi-infinite programs have been reviewed in [11, Subsection 2.3] and [19], while a hard infinite optimization problem has been recently solved via discretization in [18]. On the other hand, replacing the robust counterpart (a hard semi-infinite program when the uncertainty set is infinite) of a given uncertainty optimization problem, when it is reducible, by a finite subproblem allows many times to get the desired tractable reformulation (see e.g., [1] and [4]).

Example 2.1** (Discretizing linear infinite optimization problems)**

Consider the problems introduced in Case 4 above, with $f(\cdot):=\left\langle c^{\ast},\cdot\right\rangle$ and ${g_{t}}(x):=\left\langle a_{t}^{\ast},\cdot\right\rangle-b_{t},$ where $c^{\ast},a_{t}^{\ast}\in X^{\ast}$ and $b_{t}\in\mathbb{R},$ for all $t\in T.$ Then, $\mathrm{(RP)}_{0_{X}^{\ast}}$ collapses to the linear infinite programming problem

[TABLE]

*whose feasible set we denote by $A.$ So, $\inf\mathrm{(RP)}_{0_{X}^{\ast}}=\inf_{x\in X}\left\{\left\langle c^{\ast},x\right\rangle+i_{A}\left(x\right)\right\}.\;$ We assume that $A\neq\emptyset.$

Given $S\in U$ and ${{{\mu,\lambda}\in}}\mathbb{R}^{S},$*

[TABLE]

and

[TABLE]

Hence, $\mathrm{(ODP)}_{0_{X}^{\ast}}$ collapses to the so-called Haar dual problem [12] of $\mathrm{(RP)}_{0_{X}^{\ast}},$

[TABLE]

i.e.,

[TABLE]

From (2.8), if $\inf\mathrm{(RP)}_{0_{X}^{\ast}}=\max\mathrm{(ODP)}_{0_{X}^{\ast}}\in\mathbb{R},$ then there exist $S\in U$ and ${{{\lambda}\in}}\mathbb{R}_{+}^{S}$ such that

[TABLE]

Let $A_{S}:=\left\{x\in X:{\ \left\langle a_{s}^{\ast},x\right\rangle\leq b_{s},\;\forall s\in S}\right\}.$ Given $x\in A_{S},$ from (2.9),

[TABLE]

Since

[TABLE]

[TABLE]

*so that $\mathrm{(RP)}_{0_{X}^{\ast}}$ is reducible. Conversely, if (2.10) holds with $\inf\mathrm{(RP)}_{0_{X}^{\ast}}\in\mathbb{R}$ and $\operatorname{cone}\left\{\left({{{{{{a_{t}^{\ast},}b_{t}}}}}}\right):t\in T\right\}+\mathbb{R}_{+}\left(0_{X}^{\ast},1\right)$ is weak∗**-*closed, since $\inf\mathrm{(RP)}_{0_{X}^{\ast}}\leq\left\langle c^{\ast},x\right\rangle$ is consequence of $\left\{{\left\langle a_{s}^{\ast},x\right\rangle\leq b_{s},\;\forall s\in S}\right\},$ by the non-homogeneous Farkas lemma in lcHtvs [6] and the closedness assumption, there exist ${{{\lambda}\in}}\mathbb{R}_{+}^{S}$ and $\mu{\in}\mathbb{R}_{+}$ such that

[TABLE]

*which implies that $\mu=0$ and $\inf\mathrm{(RP)}_{0_{X}^{\ast}}=\max\mathrm{(ODP)}_{0_{X}^{\ast}}{.}$ The closedness assumption holds when $X$ is finite dimensional (guaranteeing that any finitely generated convex cone in $X^{\ast}\times\mathbb{R}$ is closed). So, as proved in [10, Theorem 8.3], a linear semi-infinite program $\mathrm{(RP)}_{0_{X}^{\ast}}$ is reducible if and only if (2.10) holds if and only if $\inf\mathrm{(RP)}_{0_{X}^{\ast}}=\max\mathrm{(ODP)}_{0_{X}^{\ast}}.$

We now assume that $\inf\mathrm{(RP)}_{0_{X}^{\ast}}=\sup\mathrm{(ODP)}_{0_{X}^{\ast}}\in\mathbb{R}.$ By (2.8), there exist sequences $\left(S_{r}\right)_{r\in\mathbb{N}}\subset U$ and $\ \left({\lambda}_{r}\right)_{r\in\mathbb{N}},$ with ${\lambda}^{r}{\in}\mathbb{R}_{+}^{S_{r}}$ for all $r\in\mathbb{N},$ such that*

[TABLE]

Denote $v_{r}:=-{{{\sum\limits_{s\in S_{r}}{{\lambda_{s}^{r}b_{s}.}}}}}$ Then,

[TABLE]

with $\lim_{r}v_{r}=\inf\mathrm{(RP)}_{0_{X}^{\ast}}.$ Let $A_{r}:=\left\{x\in X:{\ \left\langle a_{s}^{\ast},x\right\rangle\leq b_{s},\;\forall s\in S}_{r}\right\},$ $r\in\mathbb{N}.$ Given $x\in A_{r},$ from (2.11),

[TABLE]

Since $v_{r}\leq\left\langle c^{\ast},x\right\rangle$ for all $x\in A_{r},$

[TABLE]

Thus,

[TABLE]

i.e., $\mathrm{(RP)}_{0_{X}^{\ast}}$ is discretizable. Once again, the converse is true in linear semi-infinite programming [10, Corollary 8.2.1], but not in linear infinite programming.

3 Robust Conjugate Duality

We now turn back to the general perturbation function ${F_{u}}\colon X\times{Y_{u}}\rightarrow\mathbb{R}_{\infty},$ $u\in U$ and let $\Delta:=\left\{(u,y_{u}^{\ast}):u\in U,y_{u}^{\ast}\in Y_{u}^{\ast}\right\}$ be the disjoint union of the spaces $Y_{u}^{\ast}$. Recall that

[TABLE]

[TABLE]

Define $p\in\overline{\mathbb{R}}^{X}$ and $q\in\overline{\mathbb{R}}^{X^{\ast}}$ such that

[TABLE]

One then has

[TABLE]

and hence,

• •

Weak robust duality always holds:

[TABLE]

• •

Robust duality at $x^{\ast}$ means:

[TABLE]

Robust duality at $x^{\ast}$ also holds when either $p^{\ast}(x^{\ast})=+\infty$ or $q(x^{\ast})=-\infty.$

As an illustration, consider Case 4 with linear data, as in Example 2.1. Then, $p\left(x\right)=\left\langle c^{\ast},x\right\rangle+\mathrm{i}_{A}\left(x\right),$ $\operatorname{dom}p=A,$ and so

[TABLE]

Similarly, from (2.7),

[TABLE]

$\operatorname{dom}q=c^{\ast}+\operatorname{cone}\left\{a_{t}^{\ast}:t\in T\right\}$ and

[TABLE]

3.1 Basic lemmas

Let us introduce the necessary notations. Given a lcHtvs $Z$, an extended real-valued function $h\in\overline{\mathbb{R}}^{Z}$, and $\varepsilon\in\mathbb{R}_{+}$, the set of $\varepsilon$-minimizers of $h$ is defined by

[TABLE]

or, equivalently,

[TABLE]

Note that $\varepsilon-$argmin $h\neq\emptyset$ when $\inf_{Z}h\in\mathbb{R}$ and $\varepsilon>0.$ Various calculus rules involving $\varepsilon-$argmin have been given in [22].

The $\varepsilon-$subdifferential of $h$ at a point $a\in Z$ is the set (see, for instance, [15])

[TABLE]

It can be checked that if $h\in\overline{\mathbb{R}}^{X}$ is convex and $h(a)\in\mathbb{R}$, then $\partial^{\varepsilon}h(a)\neq\emptyset$ for all $\varepsilon>0$ if and only if $h$ is lower semi-continuous at $a$.

The inverse of the set-valued mapping $\partial^{\varepsilon}h:Z\rightrightarrows Z^{\ast}$ is denoted by $M^{\varepsilon}h:Z^{\ast}\rightrightarrows Z.$ For each $(\varepsilon,z^{\ast})\in\mathbb{R}_{+}\times Z^{\ast}$, we have

[TABLE]

Denoting by $\partial^{\varepsilon}h^{\ast}(z^{\ast})$ the $\varepsilon$-subdifferential of $h^{\ast}$ at $z^{\ast}\in Z^{\ast}$, namely,

[TABLE]

where $h^{\ast\ast}(z):=\sup\limits_{z^{\ast}\in Z^{\ast}}\{\langle z^{\ast},z\rangle-h^{\ast}(z^{\ast})\}$ is the biconjugate of $h$, we have

[TABLE]

with equality if and only if $h=h^{\ast\ast}.$

For each $\varepsilon\in\mathbb{R}_{+}$, we consider the set-valued mapping $S^{\varepsilon}:X^{\ast}\rightrightarrows X$ as follows:

[TABLE]

If $q(x^{\ast})=-\infty$, then $S^{\varepsilon}(x^{\ast})=p^{-1}(\mathbb{R}).$ If $q(x^{\ast})=+\infty,$ then $S^{\varepsilon}(x^{\ast})=\emptyset.$

Since $p^{\ast}\leq q$, it is clear that

[TABLE]

Lemma 3.1

Assume that $\operatorname{dom}p\neq\emptyset$. Then, for each $x^{\ast}\in X^{\ast}$, the next statements are equivalent:

$\mathrm{(i)}$* Robust duality holds at $x^{\ast}$ , i.e., $p^{\ast}(x^{\ast})=q(x^{\ast})$,*

$\mathrm{(ii)}$* $\left(M^{\varepsilon}p\right)(x^{\ast})=S^{\varepsilon}(x^{\ast}),\quad\forall\varepsilon\geq 0$,*

$\mathrm{(iii)}$* $\exists\bar{\varepsilon}>0:\left(M^{\varepsilon}p\right)(x^{\ast})=S^{\varepsilon}(x^{\ast}),\quad\forall\varepsilon\in]0,\bar{\varepsilon}[$.*

Proof. $[\mathrm{(i)}\Rightarrow\mathrm{(ii)}]$ By definition

[TABLE]

By $\mathrm{(i)}$ we thus have $\left(M^{\varepsilon}p\right)(x^{\ast})=S^{\varepsilon}(x^{\ast}).$

$[\mathrm{(ii)}\Rightarrow\mathrm{(iii)}]$ It is obviously true.

$[\mathrm{(iii)}\Rightarrow\mathrm{(i)}]$ Since $p^{\ast}(x^{\ast})\leq q(x^{\ast})$, $\mathrm{(i)}$ holds if $p^{\ast}(x^{\ast})=+\infty.$ Moreover, since $\operatorname{dom}p\neq\emptyset$, one has $p^{\ast}(x^{\ast})\neq-\infty.$ Let now $p^{\ast}(x^{\ast})\in\mathbb{R}$. In order to get a contradiction, assume that $p^{\ast}(x^{\ast})\not=q(x^{\ast})$. Then $p^{\ast}(x^{\ast})<q(x^{\ast})$ and there exists $\varepsilon\in]0,\bar{\varepsilon}[$ such that $p^{\ast}(x^{\ast})+\varepsilon<q(x^{\ast})$. Since $\inf_{x\in X}\left\{p(x)-\left\langle x^{\ast},x\right\rangle\right\}=-p^{\ast}(x^{\ast})\in\mathbb{R}$ and $\varepsilon>0,$ we have $\varepsilon-\operatorname{argmin}(p-x^{\ast})\neq\emptyset.$ Let us pick $x\in(M^{\varepsilon}p)(x^{\ast})=\varepsilon-\operatorname{argmin}(p-x^{\ast}).$ By $\mathrm{(iii)}$, we have $x\in S^{\varepsilon}(x^{\ast})$ and

[TABLE]

which contradicts $p^{\ast}(x^{\ast})+\varepsilon<q(x^{\ast})$. $\square$

For each $\varepsilon\in\mathbb{R}_{+}$, let us introduce now the the following set-valued mapping $J^{\varepsilon}:U\rightrightarrows X$:

[TABLE]

with the aim of making explicit the set $S^{\varepsilon}(x^{\ast})$. To this purpose, given $\varepsilon_{1},\varepsilon_{2}\in\mathbb{R}_{+}$, $u\in U$, and $y_{u}^{\ast}\in Y_{u}^{\ast}$, let us introduce the set-valued mapping $A_{(u,y_{u}^{\ast})}^{(\varepsilon_{1},\varepsilon_{2})}:X^{\ast}\rightrightarrows X$ such that

[TABLE]

Lemma 3.2

For each $x^{\ast}\in X^{\ast}$, $\varepsilon_{1},\varepsilon_{2}\in\mathbb{R}_{+}$, $u\in U$, and $y_{u}^{\ast}\in Y_{u}^{\ast}$, one has

[TABLE]

Proof. Let $x\in J^{\varepsilon_{1}}(u)$ be such that $(x,0_{u})\in(M^{\varepsilon_{2}}F_{u})(x^{\ast},y_{u}^{\ast}).$ Then we have** $F_{u}^{\ast}(x^{\ast},y_{u}^{\ast})\in\mathbb{R}$** and** $F_{u}(x,0_{u})\in\mathbb{R}$**. Moreover

[TABLE]

implying $p(x)\in\mathbb{R}$ and, by (3.4),

[TABLE]

that means $x\in S^{\varepsilon_{1}+\varepsilon_{2}}(x^{\ast})$ . $\square$

Lemma 3.3

Assume that

[TABLE]

Then, for each $x^{\ast}\in X^{\ast},\varepsilon\in\mathbb{R}_{+},\eta>0$, one has

[TABLE]

Proof. Let $x\in p^{-1}(\mathbb{R})$ be such that $x\in S^{\varepsilon}(x^{\ast})$, i.e.,

[TABLE]

We then have, for any $\eta>0$,

[TABLE]

and, by definition of $q$ and $p,$ there exist $u\in U$, $y_{u}^{\ast}\in Y_{u}^{\ast}$ such that

[TABLE]

Since $p(x)\in\mathbb{R}$, $F_{u}^{\ast}(x^{\ast},y_{u}^{\ast})\neq+\infty$. In fact, by (3.11), $F_{u}^{\ast}(x^{\ast},y_{u}^{\ast})\in\mathbb{R}$. Similarly, $F_{u}(x,0_{u})\in\mathbb{R}$. Setting

[TABLE]

we get $\alpha_{1}\in\mathbb{R}_{+}$, $\alpha_{2}\in\mathbb{R}.$ Actually $\alpha_{2}\geq 0$ since, by definition of conjugate,

[TABLE]

i.e., if $z=x$ and $y_{u}=0_{u},$

[TABLE]

so that

[TABLE]

Then, by (3.12), $0\leq\alpha_{1}+\alpha_{2}\leq\varepsilon+\eta$. Consequently, there exist $\varepsilon_{1},\varepsilon_{2}\in\mathbb{R}_{+}$ such that $\alpha_{1}\leq\varepsilon_{1}$, $\alpha_{2}\leq\varepsilon_{2}$, $\varepsilon_{1}+\varepsilon_{2}=\varepsilon+\eta$