One Optimization Problem with Convex Set-Valued Mapping and Duality

Abstract

This study focuses on the formulation and analysis of problems that are dual to those defined by convex set-valued mappings. Various important classes of optimization problems—such as the classical problems of mathematical and linear programming, as well as extremal problems arising in economic dynamics models—can be reduced to problems of this type. The dual problem proposed in this work is constructed on the basis of the duality theorem connecting the operations of addition and infimal convolution of convex functions, a result that has been previously applied to compact-valued mappings. It appears that, under the so-called nondegeneracy condition, this construction serves as a fundamental approach for deriving duality theorems and establishing both necessary and sufficient optimality conditions. Furthermore, alternative conditions that partially replace the nondegeneracy assumption may also prove valuable for addressing other issues within convex analysis.

1. Introduction

Convex optimization problems governed by set-valued mappings have received considerable interest in recent years owing to their sophisticated mathematical framework and importance across various domains such as nonlinear programming, operations research, optimal control theory, mathematical physics, and differential games [1,2,3,4,5,6,7,8,9,10,11,12,13]. Such problems extend the scope of classical convex optimization because they allow us to express constraints or objective functions as set-valued functions so they provide us powerful mathematical frameworks.

The article [14] introduces the concepts of set-valued mappings preserving upper and lower order. Using these concepts, some fixed-point theorems are established for partially ordered sets endowed with the hull–kernel topology for set-valued mappings. As an application of these theorems, a theorem on the existence of a solution to the vector equilibrium problem is given. In [15], a partial differential inclusion governed by a p-Laplacian containing a p-superlinear nonsmooth potential and subject to Neumann boundary conditions is studied. Using the theory of nonsmooth critical points, the existence of at least two solutions of constant sign is proved. This article [16] studies a second-order differential inclusion problem relating to a quasi-variational–hemivariational inequality with a perturbation operator in Banach spaces. The solution set of the inequality is demonstrated to be nonempty, bounded, closed, and convex through the utilization of the KKM theorem and Minty’s method. The existence of mild solutions is shown by means of techniques from fixed-point theory and weak topology.

Study [17] introduced and discussed duality operators on the set of binary extended Boolean functions. Aggregation functions have been extensively studied and applied in several practical problems involving some sort of fuzzy modeling, by enacting the fusion process of data from the unit interval. In [18], by constructing pairs of dual aggregation functions and applying them in some practical problem, one can analyze which type of behaviour of the aggregation operator can benefit the whole system. The article [19] models uncertainty in both the objective function and constraints for a robust equilibrium problem on a semi-infinite interval, including data uncertainty. A robust dual version of the problem is proposed, and weak and strong duality theorems are proved. The main topic of the paper [20] is a problem with initial conditions containing a dynamic version of the transport equation. In this problem, a delay is introduced in a function defined on a time scale, which, in turn, is used to introduce a convolution of two functions defined on a time scale. The paper [21] examines linear programming problems on time scales, which combines discrete and continuous linear programming models. Further, the weak duality theorem and the optimality conditions theorem are established and proved for arbitrary time scales, and the strong duality theorem is proved for isolated time scales.

The duality theory in optimization deals with the formulation of dual problems and the derivation of duality theorems, which are essential for analyzing the existence, uniqueness, or optimality conditions of solutions [22,23,24,25,26,27]. Although duality theory for single-valued convex optimization is thoroughly established, its extension to convex set-valued mappings presents significant challenges owing to the the complicated structure of set-valued analysis. The paper [28] uses Fenchel conjugates to establish a geometric framework for variational analysis concerning convex objects within locally convex topological spaces and Banach space contexts. The study [25] addresses the Mayer problem for third-order evolution differential inclusions using auxiliary problems with discrete and discrete-approximate inclusions. Employing Euler–Lagrange-type inclusions and transversality conditions, necessary and sufficient optimality conditions are obtained. Maximization in dual problems is realized over Euler–Lagrange-type discrete/differential inclusion solutions.

This study presents a new and comprehensive duality framework for a convex optimization problem with set-valued mapping, which extends previous results to include more general scenarios together with set-valued constraints. Utilizing the relationship between infimal convolution and convex dualization, we establish our duality theorems under substantial regularity assumptions. These results in the paper are novel and fill gaps in the literature.

The Fenchel–Rockafellar duality theorem, named after the mathematicians Werner Fenchel and R.T. Rockafellar, is perhaps the most powerful tool in all of convex analysis. The theorem arises in the study of so-called primal optimization problems $inf\left\{f\right(Ax)+g(x\left)\right\}$, where A is a bounded linear operator and $f,g$ are functions. The dual problem to the primal problem is $sup\left\{-f^{*}\left(-y^{*}\right)-g^{*}\left(A^{*}{y}^{*}\right)\right\}$, where $A^{*}$ is the dual of A. The works [22,23,24,25,26] are mainly devoted to dual problems for optimization problems described by discrete and differential inclusions. Moreover, according to Theorem 1, without the nondegeneracy condition, weak duality holds (the primal problem has an optimal value greater than or equal to the dual problem; in other words, the duality gap not less than zero). However, under the nondegeneracy condition, strong duality holds, in which the values of the primal and dual problems are equal.

The “non-degeneracy condition” plays a central role in our main results. It serves as a sufficient regularity condition, guaranteeing the absence of a duality discontinuity, a critical property for both theoretical analysis and algorithm design.

The paper is organized as follows:

Section 2 covers the basic mathematical notations and concepts used throughout the article. Fundamental results in convex analysis are recalled, key concepts related to set-valued functions are introduced, and tools such as infimal convolution and convex conjugate, which will play a central role in subsequent sections, are defined.

Section 3 describes how to construct the dual problem for the considered convex set-valued optimization problem. Using the theory of infimal convolution and convex conjugation, the dual problem is formulated, and the general duality relations that form the basis for the main results of the paper are obtained.

In Section 4, the relationship between the solutions of primal and dual problems is examined in detail.

In Section 5, the structure and properties of the solution sets for both primal and dual problems are studied. The topological and geometric properties of these sets are presented. Moreover, the connections between the solution sets under various regularity assumptions are highlighted.

Section 6 discusses additional regularity conditions that strengthen the duality framework developed in previous sections. Further duality results are presented, the flexibility of the approach is demonstrated, and its applicability to different classes of optimization problems is discussed.

Thus, drawing on fundamental results in convex analysis, we describe the construction of a dual problem for the convex optimization problem under consideration with a set-valued mapping and examine in detail the relationship between the solutions of the primal and dual problems. Along with the problems described, it is interesting to note that the obtained results can be extended to cases involving convex mathematical programming problems. The article may consider including the following promising research directions, supported by references [29,30].

2. Preliminaries

The fundamental definitions and concepts utilized in this section are primarily drawn from Mahmudov’s monograph [5]. Let X and Y denote finite-dimensional Euclidean spaces, and define $Z=X\times Y$ as their Cartesian product. For any $x\in X,y\in Y$, we write $z=(x,y)$ to represent their ordered pair, while $\langle x,y\rangle$ denotes the corresponding inner product. The elements of the dual spaces $X^{*},Y^{*}$, and $Z^{*}$ are represented by $x^{*},y^{*}$ and $z^{*}$, respectively.

Consider now a set-valued mapping $F:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^n$ which assigns to each point in ${\mathbb{R}}^n$ a subset of ${\mathbb{R}}^n$. The mapping F is convex closed if its graph is a convex and closed subset of ${\mathbb{R}}^{2n}$. The domain of F denoted by

$$domF=\{x:F(x)\ne ⌀\},$$

and consists of all points for which $F\left(x\right)$ is nonempty. Moreover, F is convex-valued if $F\left(x\right)$ is a convex set for every $x\in domF$.

Finally, let us introduce several key definitions that will be frequently employed throughout the paper:

$$\begin{array}{cc}& H_F\left(x,y^{*}\right)=\underset{y}{sup}\left\{〈y,y^{*}〉:y\in F\left(x\right)\right\},y^{*}\in {\mathbb{R}}^n,\hfill \\ & F_A\left(x;y^{*}\right)=\left\{y\in F\left(x\right):〈y,y^{*}〉=H_F\left(x,y^{*}\right)\right\}.\hfill \end{array}$$

$H_F$ and $F_A$ are called Hamiltonian function and argmaximum set for a set-valued mapping F, respectively. For convex F, we put $H_F\left(x,y^{*}\right)=-\infty$ if $F\left(x\right)=⌀$.

As usual, $W_A\left(x^{*}\right)$ is a support function of the set $A\subset {\mathbb{R}}^n$, i.e.,

$$W_A\left(x^{*}\right)=\underset{x\in A}{sup}\langle x,x^{*}\rangle ,x^{*}\in {\mathbb{R}}^n.$$

Let $\mathrm{int}A$ be the interior of the set $A\subset {\mathbb{R}}^n$ and $\mathrm{ri}A$ be the relative interior of the set A, i.e., the set of interior points of A with respect to its affine hull $\mathrm{Aff}A$.

The convex cone $K_Q\left(z\right),z=(x,y)$ is called the cone of tangent directions at a point $z\in Q$ to the set Q if from $\overline{z}=(\overline{x},\overline{y})\in K_Q\left(z\right)$, and it follows that $\overline{z}$ is a tangent vector to the set Q at point $z\in Q$, i.e., there exists such function $\eta :\mathbb{R}\to {\mathbb{R}}^{2n}$ that $z+\gamma \overline{z}+\eta \left(\gamma \right)\in Q$ for sufficiently small $\gamma >0$ and ${\gamma }^{-1}\eta \left(\gamma \right)\to 0$, as $\gamma \downarrow 0$.

A function f is called a proper function if it does not take the value $-\infty$ and is not identically equal to $+\infty$. Clearly, f is proper if and only if $domf\ne ⌀$ and $f(·)$ is finite for $x\in domf=\{x:f(x)<+\infty \}$.

In general, for a set-valued mapping F, a set-valued mapping $F^{*}:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^n$ defined by

$$F^{*}\left(y^{*};(x,y)\right):=\left\{x^{*}:\left(x^{*},-y^{*}\right)\in K_F^{*}(x,y)\right\},$$

is called the LAM to a set-valued F at a point $(x,y)\in gphF$, where $K_F^{*}(x,y)$ is the dual to the cone of tangent directions $K_{\mathrm{gphF}}(x,y)\equiv K_F(x,y)$. We provide another definition of LAM to mapping F which is more relevant for further development

$$\begin{array}{c}{F}^{*}\left(y^{*};(x,y)\right):=\left\{x^{*}:H_F\left(x_1,y^{*}\right)-H_F\left(x,y^{*}\right)\le 〈x^{*},x_1-x〉,\right.\\ \left.\forall x_1\in {\mathbb{R}}^n\right\},(x,y)\in gphF,y\in F_A\left(x;y^{*}\right).\end{array}$$

Obviously, for the convex F, the Hamiltonian function $H_F\left(·,y^{*}\right)$ is concave and the latter and previous definitions of LAMs coincide.

Definition 1.

A function $f\left(x\right)$ is said to be a closure if its epigraph epi $f=\left\{\right(x,y):y\ge f(x\left)\right\}$ is a closed set.

Definition 2.

The function $f^{*}\left(x^{*}\right)=\underset{x}{sup}\left\{〈x,x^{*}〉-f\left(x\right)\right\}$ is said to be the conjugate of f. Clearly, the conjugate function is always closed and convex.

• Let us denote

$$M_F\left(x^{*},y^{*}\right)=\underset{x,y}{inf}\left\{〈x,x^{*}〉-〈y,y^{*}〉:(x,y)\in gphF\right\}.$$

Obviously, the function

$$M_F\left(x^{*},y^{*}\right)=\underset{x}{inf}\left\{〈x,x^{*}〉-H_F\left(x,y^{*}\right)\right\}$$

is a support function taken with a minus sign.

Definition 3.

Recall that for two proper functions, $g_1$ and $g_2$, the function

$$g\left(z\right)=\underset{z_1,z_2\in Z}{inf}\left\{g_1\left(z_1\right)+g_2\left(z_2\right):z_1+z_2=z\right\}$$

is referred to as the infimal convolution of $g_1$ and $g_2$. This operation is commonly denoted by $g=g_1\oplus g_2$, $g=g_1\square g_2$, and $g=g_1▽g_2$; in what follows, we shall consistently use the notation ⊕.

It is worth noting that the operation ⊕ possesses both associative and commutative properties. Moreover, when $g_1$ and $g_2$ are proper convex closed functions, their infimal convolution $\left(g_1\oplus g_2\right)$ is itself convex and closed, although it may not necessarily remain proper.

Definition 4.

Let us recall that for a convex and closed set $C\subset {\mathbb{R}}^{2n}$, its recession (or asymptotic) cone—which is itself convex and closed (see [2,3,5])—is defined as follows:

$$0^{+}C=\{\overline{z}:z+\lambda \overline{z}\in C,\lambda >0\}.$$

We set $dom{M}_F=\left\{\left(x^{*},y^{*}\right):M_F\left(x^{*},y^{*}\right)>-\infty \right\},dom{W}_C=\left\{x^{*}:W_C\left(x^{*}\right)>-\infty \right\}$.

Definition 5.

For an arbitrary convex closed mapping F, we define, in contrast to the so-called locally adjoint mapping, what will be referred to simply as the adjoint mapping.

$$F^{*}\left(y^{*}\right)=\left\{x^{*}:\left(x^{*},-y^{*}\right)\in {\left(0^{+}gphF\right)}^{*}\right\}.$$

(1)

Let $f_0\left(x\right)$ be a convex, proper, and closed function defined on $X={\mathbb{R}}^n$, $f_0:{\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$. Let $F:{\mathbb{R}}^n\rightrightarrows {\mathbb{R}}^n$ be a convex set-valued mapping, and let P and S denote convex subsets such that $P\subseteq Y={\mathbb{R}}^n$ and $S\subseteq X={\mathbb{R}}^n$, with the property $F\left(x\right)\cap P\ne ⌀$. Our next objective is to construct the dual problem corresponding to the following optimization problem involving a set-valued mapping:

$$\begin{array}{c}\inf f_0\left(x\right),\end{array}$$

(2)

$$\begin{array}{c}F\left(x\right)\cap P\ne ⌀,\end{array}$$

(3)

$$\begin{array}{c}x\in S.\end{array}$$

(4)

3. Construction and Analysis of the Dual Problem

We replace the formulated problem (2)–(4) by the following equivalent problem in the space $Z={\mathbb{R}}^{2n}$:

$$\begin{array}{c}\inf \phi \left(z\right),\end{array}$$

(5)

$$\begin{array}{c}z\in Q.\end{array}$$

(6)

Here, $\phi \left(z\right)=f_0\left(x\right),Q=(gphF)\cap (X\times P)\cap (S\times Y)\subseteq {\mathbb{R}}^{2n}$.

Let ${\delta }_Q\left(z\right)$ denote the indicator function of the set Q. Then, the relation holds as follows:

$$\begin{array}{cc}& inf\{\phi \left(z\right):z\in Q\}=inf\{\phi \left(z\right)+{\delta }_Q\left(z\right)\}\\ & =-sup\left\{\langle z,0\rangle -\left(\phi \left(z\right)+{\delta }_Q\left(z\right)\right)\right\}\\ & =-{\left(\phi +{\delta }_Q\right)}^{*}\left(0\right)\ge -\left({\phi }^{*}\oplus {\delta }_Q^{*}\right)\left(0\right)\\ & =sup\left\{-{\phi }^{*}\left(z^{*}\right)-{\delta }_Q^{*}\left(-z^{*}\right)\right\}.\end{array}$$

In the step preceding the last equality, we employed the duality theorem that establishes the relationship between the conjugate of a sum of functions and the conjugate functions of the individual summands.

The problem:

$$sup\left\{-{\phi }^{*}\left(z^{*}\right)-{\delta }_Q^{*}\left(-z^{*}\right)\right\}$$

(7)

is said to be the dual of problems (5) and (6). If there exists a point $z\in Q$, where $\phi \left(z\right)$ is continuous, then ${\left(\phi +{\delta }_Q\right)}^{*}={\phi }^{*}\oplus {\delta }_Q^{*}$ and the values of the primal problems (5) and (6) and that of the dual problem (7) are identical. Consequently, if the infimum in problems (5) and (6) is finite, it follows that $0\in dom{\left(\phi +{\delta }_Q\right)}^{*}$ and the supremum in problem (7) exists and is achieved.

Lemma 1.

We have the relation

$${\phi }^{*}\left(z^{*}\right)=\left\{\begin{array}{cc}{f}_0^{*}\left(x^{*}\right),\hfill & y^{*}=0,\hfill \\ +\infty ,\hfill & y^{*}\ne 0.\hfill \end{array}\right.$$

Proof. 

Indeed, ${\phi }^{*}\left(z^{*}\right)=\underset{z}{sup}\left\{〈z,z^{*}〉-\phi \left(z\right)\right\}=\underset{(x,y)}{sup}\left\{〈x,x^{*}〉+〈y,y^{*}〉-f_0\left(x\right)\right\}=$$sup〈y,y^{*}〉+f_0^{*}\left(x^{*}\right)=$$\left\{\begin{array}{cc}{f}_0^{*}\left(x^{*}\right),\hfill & y^{*}=0,\hfill \\ +\infty ,\hfill & y^{*}\ne 0\hfill \end{array}\right.$ and $f_0^{*}\left(x^{*}\right)$ is also a proper closed function. Since ${\delta }_Q={\delta }_{\mathrm{gphF}}+{\delta }_{{\mathbb{R}}^n\times P}+{\delta }_{S\times {\mathbb{R}}^n}$, by the duality theorem of the operations of addition and infimal convolution of convex functions [3,5], we have

$$\begin{array}{c}{\delta }_Q^{*}\left(-z^{*}\right)\le inf\left\{{\delta }_{\mathrm{gphF}}^{*}\left(z_1^{*}\right)+{\delta }_{{\mathbb{R}}^n\times P}^{*}\left(z_2^{*}\right)+{\delta }_{S\times {\mathbb{R}}^n}^{*}\left(z_3^{*}\right):z_1^{*}+z_2^{*}+z_3^{*}+z^{*}=0\right\},\\ z_i^{*}=\left(x_i^{*},y_i^{*}\right),i=1,2,3,\end{array}$$

(8)

where, as one can easily compute,

$$\begin{array}{c}{\delta }_{\mathrm{gphF}}^{*}\left(z_1^{*}\right)=sup\left\{〈z_1,z_1^{*}〉:z_1\in gphF\right\},\\ {\delta }_{{\mathbb{R}}^n\times P}^{*}\left(z_2^{*}\right)=sup\left\{〈z_2,z_2^{*}〉:z_2\in {\mathbb{R}}^n\times P\right\}\end{array}$$

(9)

$$\begin{array}{c}=\left\{\begin{array}{cc}sup\left\{〈y_2,y_2^{*}〉:y_2\in P\right\},& x_2^{*}=0,\\ +\infty & x_2^{*}\ne 0,\end{array}\right.\\ {\delta }_{S\times {\mathbb{R}}^n}^{*}\left(z_3^{*}\right)=sup\left\{〈z_3,z_3^{*}〉:z_3\in S\times {\mathbb{R}}^n\right\}\end{array}$$

(10)

$$\begin{array}{c}=\left\{\begin{array}{cc}sup\left\{〈x_3,x_3^{*}〉:x_3\in S\right\},& y_3^{*}=0,\\ +\infty ,& y_3^{*}\ne 0.\end{array}\right.\end{array}$$

(11)

This completes the proof of the lemma. □

We shall refer to the nondegeneracy condition as being satisfied if there exists a point $z_0=\left(x_0,y_0\right)\in {\mathbb{R}}^{2n}$, such that $x_0\in ridom{f}_0$, $\left(x_0,y_0\right)\in rigphF$, $x_0\in riS$, $y_0\in riP$; alternatively, this condition also holds if $x_0\in S$, $y_0\in intP$, $\left(x_0,y_0\right)\in intgphF$ provided that $f_0\left(x\right)$ is continuous at $x_0$.

Under the nondegeneracy condition, the duality theorems mentioned above ensure that the inequality (8) holds with equality. Moreover, for every $z^{*}$ such that ${\delta }_Q^{*}\left(-z^{*}\right)<+\infty$, the infimum is achieved. Referring back to problem (7), and by invoking Lemma 1, we can directly conclude that

$$\begin{array}{cc}\hfill & {sup}_{z^{*}}\left\{-{\phi }^{*}\left(z^{*}\right)-{\delta }_Q^{*}\left(-z^{*}\right)\right\}\\ & ={sup}_{z^{*}}\left\{-f_0^{*}\left(x^{*}\right)-{\delta }_Q^{*}\left(-z^{*}\right):z^{*}=\left(x^{*},0\right),{\delta }_Q^{*}\left(-z^{*}\right)<+\infty \right\}.\end{array}$$

(12)

Now, taking into account the representations $z_1^{*}=\left(x_1^{*},y_1^{*}\right),z_2^{*}=\left(0,y_2^{*}\right),z_3^{*}=\left(x_3^{*},0\right),z^{*}=\left(x^{*},0\right)$ in the condition ${\sum }_{i=1}^3{z}_i^{*}+z^{*}=0$, we can see that the vectors $x_1^{*},x^{*},x_3^{*},y_1^{*},y_2^{*}$ must satisfy the restrictions

$$x_1^{*}+x_3^{*}+x^{*}=0,y_1^{*}+y_2^{*}=0.$$

(13)

For convenience, let us define $x^{*}=x_0^{*}, x_3^{*}=-x^{*}$ and $y_1^{*}=-y^{*}$. Incorporating these notations and the relations (13) into inequality (8), we can then state the following lemma.

Lemma 2.

For all $z^{*}=\left(x_0^{*},0\right)$ we have the inequality

$${\delta }_Q^{*}\left(-x_0^{*},0\right)\le \underset{x_0^{*},x^{*},y^{*}}{inf}\left\{-M_F\left(x_0^{*}-x^{*},-y^{*}\right)-W_P\left(-y^{*}\right)-W_S\left(x^{*}\right)\right\}.$$

Moreover, under the nondegeneracy condition, equality holds, and the infimum is attained for all $z^{*}$ in dom ${\delta }_Q^{*}$.

From this lemma and relation (12), we obtain

$$\begin{array}{cc}& sup\left\{-{\phi }^{*}\left(z^{*}\right)-{\delta }_Q^{*}\left(-z^{*}\right)\right\}\\ & \ge {sup}_{x_0^{*},x^{*},y^{*}}\left\{-f_0^{*}\left(x_0^{*}\right)+M_F\left(x_0^{*}-x^{*},-y^{*}\right)+W_P\left(-y^{*}\right)+W_S\left(x^{*}\right)\right\}.\end{array}$$

The constructed problem:

$$sup\left\{-f_0^{*}\left(x_0^{*}\right)+M_F\left(x_0^{*}-x^{*},-y^{*}\right)+W_P\left(-y^{*}\right)+W_S\left(x^{*}\right)\right\}$$

(14)

and it is called the dual of problem (2)–(4). Taking all these facts into account, the result can be succinctly formulated as the following theorem.

Theorem 1.

Denote by α and ${\alpha }^{*}$ the values of the primal and dual problems, respectively. It always holds that $\alpha \ge {\alpha }^{*}$. Moreover, under the nondegeneracy condition, if one of the problems admits a solution, the other does as well, and their optimal values are equal. In the case where $\alpha >-\infty$, a solution to the dual problem exists.

Remark 1.

In problems where $f_0\left(y\right)$ replaces $f_0\left(x\right)$, one can similarly formulate the corresponding dual problem as follows:

$$sup\left\{-f_0^{*}\left(y^{*}\right)+M_F\left(-x^{*},-y_1^{*}\right)+W_P\left(y^{*}-y_1^{*}\right)+W_S\left(x^{*}\right)\right\}.$$

We now turn our attention to the linear programming problem:

$$inf\{\langle c,x\rangle :Ax\le b\},$$

(15)

where A is an $m\times n$ matrix, $c\in {\mathbb{R}}^n,b\in {\mathbb{R}}^m$.

By employing the infimal convolution, we attempt to formulate the dual of problem (15). Note that $\phi \left(x\right)=\langle c,x\rangle$ is linear and $Q=\{x:Ax\le b\}$ is a polyhedral set. Consequently, Theorem 20.1.6 from [3] can be applied without requiring the nondegeneracy condition.

$$inf\{\phi \left(x\right):x\in Q\}=sup\left\{-{\phi }^{*}\left(x^{*}\right)-{\delta }_Q^{*}\left(-x^{*}\right)\right\}.$$

(16)

On one hand, it is clear that

$${\phi }^{*}\left(x^{*}\right)=\underset{x}{sup}〈x,x^{*}-c〉=\left\{\begin{array}{cc}0,\hfill & x^{*}=c,\hfill \\ +\infty ,\hfill & x^{*}\ne c.\hfill \end{array}\right.$$

On the other hand,

$$\begin{array}{c}{\delta }_Q^{*}\left(-x^{*}\right)=sup\left\{〈x,-x^{*}〉-{\delta }_Q\left(x\right)\right\}=\underset{x}{sup}\{〈x,-x^{*}〉-\underset{y^{*}\ge 0}{sup}〈y^{*},Ax-b〉\}\\ =\underset{x}{sup}\underset{y^{*}\ge 0}{inf}\left\{〈x,-x^{*}-A^{*}{y}^{*}〉+〈y^{*},b〉\right\}.\end{array}$$

Thus,

$${\delta }_Q^{*}(-x^{*})=\left\{\begin{array}{cc}\underset{y^{*}\ge 0}{inf}\langle y^{*},b\rangle ,\hfill & x^{*}+A^{*}{y}^{*}=0,\hfill \\ +\infty ,\hfill & x^{*}+A^{*}{y}^{*}\ne 0.\hfill \end{array}\right.$$

By Theorem 6.5.2 of [31], we have $dom\left({\phi }^{*}\oplus {\delta }_Q^{*}\right)=dom{\phi }^{*}+dom{\delta }_Q^{*}$ and, therefore, if $0\in dom({\phi }^{*}\oplus {\delta }_Q^{*})$, i.e., $c+A^{*}{y}^{*}=0$, then

$$sup\left\{-{\phi }^{*}\left(x^{*}\right)-{\delta }_Q^{*}\left(-x^{*}\right)\right\}=-{\delta }_Q^{*}(-c)=\underset{y^{*}\ge 0}{sup}\left\{〈-y^{*},b〉:c+A^{*}{y}^{*}=0\right\}.$$

Thus, finally, relation (16) has the form

$$inf\{\langle c,x\rangle :Ax\le b\}=\underset{y^{*}\ge 0}{sup}\left\{〈-y^{*},b〉:A^{*}{y}^{*}+c=0\right\}.$$

Note that the duality result for linear programming has an economic interpretation; if we interpret the primal problem as a classical “resource allocation” problem, then its dual problem can be interpreted as a “resource pricing” problem.

4. Correspondence Between the Solutions of the Primal and Dual Problems

The following Theorems 2 and 3 show how the solutions of the primary and dual problems are related. Furthermore, Theorem 2 provides necessary and sufficient optimality conditions for problem (2)–(4).

Theorem 2.

Under the nondegeneracy condition, we proceed as follows. Then, in order that the point $\tilde{x}$ be a solution of the problem (2)–(4), it is necessary and sufficient that there exist vectors $x_0^{*}\in \partial f_0\left(\tilde{x}\right)$, $-y^{*}\in K_P^{*}\left(\tilde{y}\right)$, $x^{*}\in K_S^{*}\left(\tilde{x}\right)$ not equal to zero simultaneously, such that $x^{*}-x_0^{*}\in F^{*}(y^{*};\tilde{z}),\tilde{z}=(\tilde{x},\tilde{y}),\tilde{y}\in F\left(\tilde{x}\right)\cap P$.

Proof. 

The proof is analogous to the proof of Corollary 3.8 [5]. The difference is that under the regularity condition in the convex case, $\lambda =1$ in Corollary 3.8. Indeed, taking

$$A_0=dom{f}_0\times {\mathbb{R}}^n,A_1=gphF,A_2={\mathbb{R}}^n\times P,A_3=S\times {\mathbb{R}}^n,$$

we observe that either

$$(int{A}_0)\cap (int{A}_1)\cap (int{A}_2)\cap A_3\ne ⌀$$

or

$$(ri{A}_0)\cap (ri{A}_1)\cap (ri{A}_2)\cap (ri{A}_3)\ne ⌀.$$

Therefore, taking into account these conditions, by Theorem 3.3 and 3.4 [5], we get the desired result. □

Theorem 3.

Assuming that $\tilde{x}$ solves problem (2)–(4) and the nondegeneracy condition holds, the collection of vectors $(x_0^{*},x^{*},y^{*})$ forms a solution of the dual problem (14) if, and only if, they satisfy the necessary and sufficient conditions outlined in Theorem 2.

Proof. 

Let $(x_0^{*},x^{*},y^{*})$ be a solution of the dual problem. Since the problems (2)–(6) are equivalent, from $\tilde{z}=(\tilde{x},\tilde{y})\in \left\{z:\phi \left(z\right)+{\delta }_Q\left(z\right)=\alpha \right\}$ there follows that $0\in \partial (\phi \left(\tilde{z}\right)+{\delta }_Q\left(\tilde{z}\right))$ or $\tilde{z}\in \partial {(\phi +{\delta }_Q)}^{*}\left(0\right)=\partial ({\phi }^{*}\oplus {\delta }_Q^{*})\left(0\right)$. However, by the assumption of the theorem and by Theorem 6.6.6 of [31], we have $\partial ({\phi }^{*}\oplus {\delta }_Q^{*})\left(0\right)=\partial {\phi }^{*}\left(z^{*}\right)\cap \partial {\delta }_Q^{*}(-z^{*})$. Thus,

$$\tilde{z}\in \partial {\phi }^{*}\left(z^{*}\right)\cap \partial {\delta }_Q^{*}(-z^{*}),$$

(17)

where $z^{*}$ is a solution of the dual problem (7). Next, by the assumption of the theorem, the vectors $z^{*}=(x_0^{*},0),z_1^{*}=(x^{*}-x_0^{*},-y^{*}),z_2^{*}=(0,y^{*}),z_3^{*}=(-x^{*},0)({\sum }_{i=1}^3{z}_i^{*}+z^{*}=0)$ supply a minimum in formula (8) and exact equality

$${\delta }_Q^{*}(-z^{*})={\delta }_{\mathrm{gphF}}^{*}\left(z_1^{*}\right)+{\delta }_{{\mathbb{R}}^n\times P}^{*}\left(z_2^{*}\right)+{\delta }_{S\times {\mathbb{R}}^n}^{*}\left(z_3^{*}\right)$$

is attained. Then, from formula (17), there follows that

$$\tilde{z}\in \partial {\phi }^{*}\left(z^{*}\right)\cap \partial {\delta }_{gphF}^{*}\left(z_1^{*}\right)\cap \partial {\delta }_{{\mathbb{R}}^n\times P}^{*}\left(z_2^{*}\right)\cap \partial {\delta }_{S\times {\mathbb{R}}^n}^{*}\left(z_3^{*}\right).$$

(18)

From here (18), we obtain the conditions of Theorem 2. Really, $\tilde{z}\in \partial {\phi }^{*}\left(z^{*}\right)$ means that $z^{*}\in \partial \phi \left(\tilde{z}\right)$ or $\left(x_0^{*},0\right)\in \left(\partial f_0\left(\tilde{x}\right)\times 0\right)$, i.e., $x_0^{*}\in \partial f_0\left(\tilde{x}\right)$. Further, $\tilde{z}\in \partial {\delta }_{\mathrm{gphF}}^{*}\left(z_1^{*}\right)$ is equivalent to the inclusion $\left(x^{*}-x_0^{*}\right.$, $\left.-y^{*}\right)\in \partial {\delta }_{\mathrm{gphF}}\left(\tilde{z}\right)=-K_F^{*}\left(\tilde{z}\right)$ or, by the definition of a locally adjoint mapping, $x^{*}-x_0^{*}\in F^{*}(y^{*};\tilde{z})$. Similarly, from the inclusions $\tilde{z}\in \partial {\delta }_{{\mathbb{R}}^n\times P}^{*}\left(z_2^{*}\right)$ and $\tilde{z}\in \partial {\delta }_{S\times {\mathbb{R}}^n}^{*}\left(z_3^{*}\right)$, there follows that $-y^{*}\in K_P^{*}\left(\tilde{y}\right)$ and $x^{*}\in K_S^{*}\left(\tilde{x}\right)$.

Now, we prove the inverse inclusion: assume that the collection $(x_0^{*},x^{*},y^{*})$ satisfies the conditions of Theorem 2. It is known [3] that the inclusion $x_0^{*}\in \partial f_0\left(\tilde{x}\right)$ is equivalent to the condition

$$f_0^{*}\left(x_0^{*}\right)=〈\tilde{x},x_0^{*}〉-f_0\left(\tilde{x}\right).$$

(19)

Further, the inclusion $x^{*}-x_0^{*}\in F^{*}(y^{*};\tilde{z})$ is equivalent to the relation

$$M_F\left(x_0^{*}-x^{*},-y^{*}\right)+〈\tilde{x},x^{*}-x_0^{*}〉=-H_F\left(\tilde{x},-y^{*}\right).$$

(20)

The inclusions $x^{*}\in K_S^{*}\left(\tilde{x}\right)$ and $-y^{*}\in K_P^{*}\left(\tilde{y}\right)$ mean that $W_S\left(x^{*}\right)=〈x^{*},\tilde{x}〉$ and $-〈\tilde{y},y^{*}〉=W_P\left(-y^{*}\right)$, respectively. From this, and also from the relations (19) and (20), taking into account $H_F(\tilde{x},-y^{*})=〈-y^{*},\tilde{y}〉$, we obtain that

$$\begin{array}{c}-f_0^{*}\left(x_0^{*}\right)+M_F\left(x_0^{*}-x^{*},-y^{*}\right)+W_P\left(-y^{*}\right)+W_S\left(x^{*}\right)=\\ =-〈\tilde{x},x_0^{*}〉+f_0\left(\tilde{x}\right)+〈\tilde{y},y^{*}〉-〈\tilde{x},x^{*}-x_0^{*}〉-〈y^{*},\tilde{y}〉+〈x^{*},\tilde{x}〉=f_0\left(\tilde{x}\right).\end{array}$$

Consequently, ${\alpha }^{*}\ge \alpha$, which together with $\alpha \ge {\alpha }^{*}$, yields $\alpha ={\alpha }^{*}$. This concludes the proof of the theorem. □

5. Representation of Solution Sets

For the present section, we consider $\phi \left(z\right):{\mathbb{R}}^{2n}\to \mathbb{R}\cup \{+\infty \}$ as an arbitrary, convex, proper, and closed function, with $Q\subseteq {\mathbb{R}}^{2n}$ a convex set. Denote by $W_Q\left(z^{*}\right),z^{*}\in Z^{*}={\mathbb{R}}^{2n}$ the support function of Q, and let A and $A^{*}$ be the solution sets of the primal and dual problems, respectively, as follows:

$$\begin{array}{c}A=\{z\in Q:\phi (z)=\alpha \},\\ A^{*}=\left\{z^{*}:-{\phi }^{*}\left(z^{*}\right)-{\delta }_Q^{*}\left(-z^{*}\right)={\alpha }^{*}\right\}.\end{array}$$

As before, $\alpha$ and ${\alpha }^{*}$ represent the values of the primal and dual problems, respectively.

Theorem 4.

Suppose there exists a point $z_1\in Q$ where $\phi \left(z\right)$ is continuous, and let α be a finite value. Then, the solution sets $A^{*}$ and A of the dual and primal problems, respectively, take the form

$$\begin{array}{c}{A}^{*}=\left(\partial \phi \left(z\right)\cap \partial {\delta }_Q(-z)\right),z\in A;\\ A=\left(\partial {\phi }^{*}\left(z^{*}\right)\cap \partial {\delta }_Q^{*}\left(-z^{*}\right)\right),z^{*}\in A^{*}.\end{array}$$

Proof. 

We recall that, by the assumption of the theorem, we have $inf\{\phi \left(z\right):z\in Q\}=sup\{-{\phi }^{*}\left(z^{*}\right)-{\delta }_Q^{*}\left(-z^{*}\right)\}$, if $z\in A$, then $\partial \phi \left(z\right)\cap K_Q^{*}\left(z\right)\ne ⌀$. However, since $K_Q^{*}\left(z\right)=-\partial {\delta }_Q\left(z\right)=\partial {\delta }_Q(-z)$, then $z\in A$ if and only if $\partial \phi \left(z\right)\cap \partial {\delta }_Q(-z)\ne ⌀$. Then, applying Theorems 6.6.5 and 6.6.6 of [31], we obtain

$$\partial \phi \left(z\right)\cap \partial {\delta }_Q(-z)=\partial \left(\phi \oplus {\delta }_Q\right)\left(0\right)\ne ⌀.$$

(21)

That is why, from the relation ${({\phi }^{*}+{\delta }_Q^{*})}^{*}\left(0\right)=(\phi \oplus {\delta }_Q)\left(0\right)$ and from formula (21), there follows that $\partial \phi \left(z\right)\cap \partial {\delta }_Q(-z)=\partial {({\phi }^{*}+{\delta }_Q^{*})}^{*}\left(0\right)$.

Thus, $z^{*}\in \partial \phi \left(z\right)\cap \partial {\delta }_Q(-z)$ if and only if $z^{*}\in \partial {({\phi }^{*}+{\delta }_Q^{*})}^{*}\left(0\right)$ or $0\in \partial \left({\phi }^{*}(z^{*}\right)+{\delta }_Q^{*}(-z^{*}))$.

This last inclusion shows that $z^{*}$ solves the dual problem, from which it is straightforward to conclude that

$$A^{*}=\left(\partial \phi \left(z\right)\cap \partial {\delta }_Q(-z)\right),z\in A$$

From here, by a simple standard verification, we can derive the required structure of the set $A^{*}$.

Similarly, using formula (17) together with the equality ${\delta }_Q^{*}\left(-z^{*}\right)=W_Q\left(-z^{*}\right)$, the form of the solution set A can be explicitly determined. □

Remark 2.

If $\phi \left(z\right)=f_0\left(x\right)$ and Q is defined by the constraint of the problem (2)–(4), then, under the nondegeneracy condition, we obtain that

$$K_Q^{*}\left(\tilde{z}\right)=K_{\mathrm{gphF}}^{*}\left(\tilde{z}\right)+K_{{\mathbb{R}}^n\times P}^{*}\left(\tilde{z}\right)+K_{S\times {\mathbb{R}}^n}^{*}\left(\tilde{z}\right).$$

Knowing that $K_{{\mathbb{R}}^n\times P}^{*}\left(\tilde{z}\right)=\left\{0\right\}\times K_P^{*}\left(\tilde{y}\right)$, $K_{S\times {\mathbb{R}}^n}^{*}\left(\tilde{z}\right)=K_S^{*}\left(\tilde{x}\right)\times \left\{0\right\}$, and $\partial \phi \left(z\right)=\partial f_0\left(x\right)\times \left\{0\right\}$, and writing $z^{*}\in \partial \phi \left(\tilde{z}\right)\cap K_Q^{*}\left(\tilde{z}\right)$, one can conclude—as in Theorem 3—that $z^{*}\in A^{*}$ if and only if there exists a uniquely determined collection $(x^{*},x_0^{*},y^{*})$ that satisfies the conditions of Theorem 2.

6. Additional Conditions and Duality

Our next goal is to investigate duality properties under certain convex analysis conditions. For this purpose, we introduce the following auxiliary lemmas.

Lemma 3.

The following inclusion holds:

$$\begin{array}{cc}& dom({\phi }^{*}\oplus {\delta }_Q^{*})\supseteq \left\{\right(x_1^{*}+x_3^{*}+x^{*},\\ & y_1^{*}+y_2^{*}):M_F\left(-x_1^{*},y_1^{*}\right)>-\infty ,\\ & W_P(-y_2^{*})>-\infty ,W_S(-x_3^{*})>-\infty ,x^{*}\in dom{f}_0^{*}\},\end{array}$$

and under the nondegeneracy condition, these sets coincide exactly.

Proof. 

By Theorem 6.5.2 of [31], we have $dom({\phi }^{*}\oplus {\delta }_Q^{*})=dom{\phi }^{*}+dom{\delta }_Q^{*}$, while $dom{\delta }_Q^{*}\supseteq dom{\delta }_{\mathrm{gphF}}^{*}+dom{\delta }_{{\mathbb{R}}^n\times P}^{*}+dom{\delta }_{S\times {\mathbb{R}}^n}^{*}$, where, under the condition of nondegeneracy, we have strict equality. Then, taking into account that $dom{\phi }^{*}=\left\{\left(x^{*},0\right):x^{*}\in dom{f}_0^{*}\right\}$ and making use of the relations (9)–(11), we conclude the proof of the lemma. □

Lemma 4.

Given convex closed mappings F and convex sets C, we have $dom{M}_F\subseteq {(0^{+}gphF)}^{*}$, $dom{W}_C\subseteq {\left(0^{+}C\right)}^{*}$.

If $gphF=gph{F}_1+gph{F}_2,C=M+K$, where $gph{F}_1$ and M are bounded closed sets, while $gph{F}_2$ and K are convex closed cones, then

$$dom{M}_F={(0^{+}gph{F}_2)}^{*},dom{W}_C=K^{*}.$$

Proof. 

We prove the statement for the mapping F noting that the argument for the sets C proceeds in a similar manner. Consider $(x_0^{*},-y_0^{*})\in dom{M}_F$. One has to prove that $(x_0^{*},-y_0^{*})\in {(0^{+}gphF)}^{*}$. We assume the opposite. Then, there exists a pair $({\overline{x}}_0,{\overline{y}}_0)\in 0^{+}gphF$, such that $\langle {\overline{x}}_0,x_0^{*}\rangle -\langle {\overline{y}}_0,y_0^{*}\rangle <0$. By definition, $0^{+}gphF+\lambda ({\overline{x}}_0,{\overline{y}}_0)\subset gphF,(x,y)\in gphF,\lambda >0$. Then, $\langle x+\lambda {\overline{x}}_0,x_0^{*}\rangle +\langle y+\lambda {\overline{y}}_0,y_0^{*}\rangle =\langle x_0^{*},x\rangle -\langle y_0^{*},y\rangle +\lambda \{\langle {\overline{x}}_0,x_0^{*}\rangle -\langle {\overline{y}}_0,y_0^{*}\rangle \}\to -\infty$, as $\lambda \to +\infty$. The contradiction above proves the first statement of the lemma. Consider, for instance, the parabolic mapping $(F\left(x\right)=\{y:y\ge x^2,x,y\in \mathbb{R}\})$ which illustrates that the inverse inclusion generally fails. Nevertheless, if $gphF=gph{F}_1+gph{F}_2$, then by Corollary 9.1.2 from [3], we can write

$$0^{+}(gph{F}_1+gph{F}_2)=0^{+}gph{F}_1+0^{+}gph{F}_2=0^{+}gph{F}_2.$$

In addition,

$$\begin{array}{cc}\hfill dom& M_F=dom\left(M_{F_1}+M_{F_2}\right)=dom{M}_{F_1}\cap dom{M}_{F_2}\\ & =Z^{*}\cap dom{M}_{F_2}=dom{M}_{F_2}.\end{array}$$

(22)

Further, it is clear that

$$M_{F_2}\left(x^{*},-y^{*}\right)=\left\{\begin{array}{cc}0,\hfill & \left(x^{*},-y^{*}\right)\in {(0^{+}gph{F}_2)}^{*},\hfill \\ -\infty ,\hfill & \left(x^{*},-y^{*}\right)\notin {(0^{+}gph{F}_2)}^{*}\hfill \end{array}\right.$$

and, therefore, $dom{M}_{F_2}={(0^{+}gph{F}_2)}^{*}$. Taking into account relation (22), we have $dom{M}_F={(0^{+}gph{F}_2)}^{*}$. The lemma is proved. □

Theorem 5.

Suppose that the mapping F and the sets P and S are convex and closed, and that the nondegeneracy condition is satisfied. Then, for the dual problem (14), the value ${\alpha }^{*}$ is finite and attainable if and only if—under the representability condition of Lemma 4—it holds the following:

$$x_0^{*}\in dom{f}_0^{*},x^{*}-x_0^{*}\in F^{*}\left(y^{*}\right),-y^{*}\in {\left(0^{+}P\right)}^{*},x^{*}\in {\left(0^{+}S\right)}^{*}.$$

Proof. 

Suppose that ${\alpha }^{*}$ is finite and achieved. Then, by the arguments in the proof of Theorem 1, we have $\alpha ={\alpha }^{*}$ and additionally,

$$0\in dom{\left(\phi +{\delta }_Q\right)}^{*}=dom\left({\phi }^{*}\oplus {\delta }_Q^{*}\right).$$

By Lemma 3, this means that $x_1^{*}+x_3^{*}+x^{*}=0$, $y_1^{*}+y_2^{*}=0$, $M_F\left(x_1^{*},-y_1^{*}\right)>-\infty$, $W_P\left(-y_2^{*}\right)>-\infty$, $W_S\left(-x_3^{*}\right)>-\infty$, $x^{*}\in dom{f}_0^{*}$ or (with the previous notations) $\left(x_0^{*}-x^{*},y^{*}\right)\in dom{M}_F$, $-y^{*}\in dom{W}_P$, $x^{*}\in dom{W}_S$, $x_0^{*}\in dom{f}_0^{*}$. Using the statement from the first half of Lemma 4, it follows that

$$\left(x^{*}-x_0^{*},-y^{*}\right)\in {(0^{+}gphF)}^{*},-y^{*}\in {\left(0^{+}P\right)}^{*},x^{*}\in {\left(0^{+}S\right)}^{*}.$$

Considering the first inclusion and using Definition 5, we obtain

$$x^{*}-x_0^{*}\in F^{*}\left(y^{*}\right).$$

The aforementioned relations, together with the inclusion $x_0^{*}\in dom{f}_0^{*}$, establish the necessity of the stated condition.

Let us now assume that the condition specified in Lemma 4 concerning the representability of the mapping F, as well as of the sets P and S is satisfied. Consequently, by virtue of the same lemma and the hypotheses of the theorem, it follows that

$$\begin{array}{c}dom{M}_F={(0^{+}gphF)}^{*},dom{W}_P={\left(0^{+}P\right)}^{*},dom{W}_S={\left(0^{+}S\right)}^{*},\end{array}$$

where ${(0^{+}gphF)}^{*}=\left\{\left(x^{*}-x_0^{*},-y^{*}\right):x^{*}-x_0^{*}\in F^{*}\left(y^{*}\right)\right\}$. Therefore, by tracing the argument in the reverse order, one can verify the sufficiency of the conditions enumerated in the theorem. □

The subsequent theorem provides a sufficient condition, expressed through the recession function $f_0^{*}{0}^{+}$ associated with $f_0^{*}$, under which the primal and dual problem values coincide. It is worth recalling that the recession function $f_0^{*}{0}^{+}$ is defined by virtue of the relation

$$epi\left(f_0^{*}{0}^{+}\right)=0^{+}\left(\mathrm{epi}{f}_0^{*}\right).$$

Theorem 6.

A sufficient condition for the equivalence of the optimal values of the primal problem (2)–(4) and the dual problem (14) is that there does not exist any collection of vectors $x^{*},y^{*},x_0^{*}$ that constitute a solution to the inequality

$$\left(f_0^{*}{0}^{+}\right)\left(x_0^{*}\right)-M_F\left(x_0^{*}-x^{*},-y^{*}\right)-W_P\left(-y^{*}\right)-W_S\left(x^{*}\right)\le 0.$$

Proof. 

It is straightforward to verify, for the function ${\phi }^{*}$ (see Lemma 1), that the recession cone of its epigraph satisfies

$$0^{+}\left(\mathrm{epi}{\phi }^{*}\right)=\left\{\left({\overline{z}}^{*},v\right):\left({\overline{x}}^{*},v\right)\in 0^{+}\left(\mathrm{epi}{f}_0^{*}\right),{\overline{y}}^{*}=0,v\in \mathbb{R}\right\}.$$

Then, obviously, the recession function of the function ${\phi }^{*}$ will have the form

$$\left({\phi }^{*}{0}^{+}\right)\left({\overline{z}}^{*}\right)=\left(f_0^{*}{0}^{+}\right)\left({\overline{x}}^{*}\right),{\overline{y}}^{*}=0.$$

(23)

Moreover, due to the positive homogeneity of the conjugate functions associated with the indicator functions, we obtain that

$$\begin{array}{c}\left({\delta }_{\mathrm{gphF}}^{*}{0}^{+}\right)\left(z_1^{*}\right)={\delta }_{\mathrm{gphF}}^{*}\left(z_1^{*}\right),\\ \left({\delta }_{{\mathbb{R}}^n\times P}^{*}{0}^{+}\right)\left(z_2^{*}\right)={\delta }_P^{*}\left(y_2^{*}\right),x_2^{*}=0,\\ \left({\delta }_{S\times {\mathbb{R}}^n}^{*}{0}^{+}\right)\left(z_3^{*}\right)={\delta }_S^{*}\left(x_3^{*}\right),y_3^{*}=0.\end{array}$$

(24)

From the relations (23) and (24), by virtue of Corollary 16.2.2 of [31], we can assert that if the inequality

$$\left(f_0^{*}{0}^{+}\right)\left({\overline{x}}_0^{*}\right)+{\delta }_{\mathrm{gphF}}^{*}\left(z_1^{*}\right)+{\delta }_S^{*}\left(x_3^{*}\right)+{\delta }_P^{*}\left(y_2^{*}\right)\le 0$$

(25)

does not have solutions under the conditions ${\overline{x}}_0^{*}+x_1^{*}+x_3^{*}=0$, $y_1^{*}+y_2^{*}=0$, then $ri\left(dom\phi \cap dom{\delta }_Q\right)\ne ⌀$.

At the same time, setting $A=dom{f}_0\times {\mathbb{R}}^n,A_1=gphF,A_2={\mathbb{R}}^n\times P$ and $A_3=S\times {\mathbb{R}}^n$, it is easy to see that the existence of points $x^0\in S$, $y^0\in intP$, $(x^0,y^0)\in intgphF$ such that $f_0$ is continuous at the point $x^0$, means that

$$(intA)\cap \left(int{A}_1\right)\cap \left(int{A}_2\right)\cap A_3\ne ⌀.$$

(26)

Provided that there are no vectors $z_1^{*},z_2^{*},z_3^{*},z^{*}$, fulfilling the specified constraint

$${\delta }_A^{*}\left(z^{*}\right)+{\delta }_{A_1}^{*}\left(z_1^{*}\right)+{\delta }_{A_2}^{*}\left(z_2^{*}\right)+{\delta }_{A_3}^{*}\left(z_3^{*}\right)\le 0$$

(27)

and such that $z_1^{*}+z_2^{*}+z_3^{*}+z^{*}=0$, then condition (26) is satisfied. Further, it is easy to compute that

$${\delta }_A^{*}\left(z^{*}\right)=\left\{\begin{array}{cc}{\delta }_{\mathrm{dom}{f}_0}^{*}\left(x^{*}\right),& y^{*}=0,\\ +\infty ,& y^{*}\ne 0.\end{array}\right.$$

By applying Theorem 6.8.5 from [31], it can be readily verified that $\left(f_0^{*}{0}^{+}\right)\left({\overline{x}}_0^{*}\right)={\delta }_{\mathrm{dom}{f}_0}^{*}\left({\overline{x}}_0^{*}\right)$. It follows immediately that inequality (27) coincides with inequality (25). Hence, under the conditions ${\overline{x}}_0^{*}+x_1^{*}+x_3^{*}=0$, $y_1^{*}+y_2^{*}=0$, the inconsistency of inequality (25) ensures that the nondegeneracy condition is satisfied. Consequently, reverting to the earlier notation in inequality (25), and invoking Theorem 1, we arrive at the desired result. □