Two Optimal Value Functions in Parametric Conic Linear Programming¹¹1Dedicated to Professor Franco Giannessi on the occasion of his 85th birthday.

If the feasible region or the objective function of a mathematical programming problem depends on a parameter, then the optimal value of the problem is a function of the parameter. In general, the optimal value function is a fairly complicated function. Continuity properties (resp., differentiability properties) of the optimal value function in parametric mathematical programming are usually classified as results on stability (resp., on differential stability) of optimization problems. Many results in this direction can be found in the books (Bonnans_Shapiro_2000, , Chapters 4,5), (Mordukhovich_Book_I, , Chapter 4) and (Mordukhovich_Book_II, , Chapter 5), the papers [4-13], the dissertation An_Thesis , and the references therein.

In 2001, Gauvin Gauvin_2001 investigated the sensitivity of the optimal value of a parametric linear programming problem. He gave separated formulas for the increment of the optimal value with respect to perturbations of the right-hand-side vector in the constraint system, the cost vector, and the coefficients of the matrix defining the linear constraint system. In 2016, Thieu Thieu_2016 established some lower and upper bounds for the increment of the optimal value of a parametric linear program, where both objective vector and right-hand-side vector in the constraint system are subject to perturbations. He showed that the appearance of an extra second-order term in these estimates is a must.

Conic linear programming is a natural extension of linear programming. In a linear program, a feasible point usually has nonnegative components, i.e., it belongs to the nonnegative orthant of an Euclidean space. In addition, the constraints are written as linear inequalities. Meanwhile, in a conic linear program, the constraint is written as an inequality with the ordering cone being a closed convex cone in an Euclidean space. Conic linear programs can be used in many applications that cannot be simulated by linear programs (see, e.g., (BN_2001, , Section 2.2) and (Luenberger_Ye_2016, , Chapter 6)). This is the reason why these special convex optimization problems have attracted much attention from researchers (see Ben-Tal and Nemirovski BN_2001 , Bonnans and Shapiro Bonnans_Shapiro_2000 , Luenberger and Ye Luenberger_Ye_2016 , Chuong and Jeyakumar Chuong_Jeyakumar_2017 , and the references therein).

It is of interest to know whether results similar to those of Gauvin Gauvin_2001 and Thieu Thieu_2016 can be obtained for conic linear programs, or not. The aim of this paper is to show that the results of Gauvin_2001 admit certain generalizations in conic linear programming, and some differentiability properties, as well as a locally Lipschitz continuity property, can be established for the two related optimal value functions. The obtained results are analyzed via four concrete examples which show that the optimal value functions in a conic linear program are more complicated than that of the corresponding optimal value functions in linear programming.

The remaining part of our paper has six sections. Section 2 contains some definitions, a strong duality theorem in conic linear programming, and a lemma. Section 3 is devoted to the locally Lipschitz continuity property of the optimal value function of a conic linear program under right-hand-side perturbations. Differentiability properties of this optimal value function are studied in Section 4. Some increment formulas for the function are established in Section 5. The optimal value function of a conic linear program under linear perturbations of the objective function is studied in Section 6. The final section presents some concluding remarks.

One says that a nonempty subset $K$ of the Euclidean space $\mathbb{R}^{m}$ is a cone if $tK\subset K$ for all $t\geq 0$. The (positive) dual cone $K^{*}$ of a cone $K\subset\mathbb{R}^{m}$ is given by $K^{*}=\{v\in\mathbb{R}^{m}\;:\;v^{T}y\geq 0\quad\forall y\in K\}$, where ^T denotes the matrix transposition. Herein, vectors of Euclidean spaces are written as rows of real numbers in the text, but they are interpreted as columns of real numbers in matrix calculations. The closed ball centered at $a\in\mathbb{R}^{m}$ with radius $\varepsilon>0$ is denoted by $\bar{B}(a,\varepsilon)$. By $\overline{\mathbb{R}}$ we denote the set $\mathbb{R}\cup\{\pm\infty\}$ of extended real values.

Let $f:\mathbb{R}^{k}\to\overline{\mathbb{R}}$ be a proper convex function and $\bar{x}\in\mathbb{R}^{k}$ be such that $f(\bar{x})$ is finite. According to Theorem 23.1 in Rockafellar_1970 , the directional derivative

of $f$ at $\bar{x}$ w.r.t. a direction $h\in\mathbb{R}^{k}$, always exists (it can take values $-\infty$ or $+\infty$), and one has $f^{\prime}(\bar{x};h)=\inf\limits_{t>0}\dfrac{f(\bar{x}+th)-f(\bar{x})}{t}.$ The subdifferential of $f$ at $\bar{x}$ is defined by $\partial f(\bar{x})=\big{\{}p\in\mathbb{R}^{k}\;:\;p^{T}(x-\bar{x})\leq f(x)-f(\bar{x})\ \,\forall x\in\mathbb{R}^{k}\big{\}}.$

For a convex set $C$ in an Euclidean space $\mathbb{R}^{k}$, the normal cone of $C$ at $\bar{x}\in C$ is given by $N(\bar{x};C)=\left\{p\in\mathbb{R}^{k}\;:\;p^{T}(x-\bar{x})\leq 0\ \,\forall x\in C\right\}.$

Let $F:\mathbb{R}^{k}\rightrightarrows\mathbb{R}^{\ell}$ be a multifunction. The domain and the graph of $F$ are given, respectively, by the formulas ${\rm{dom}}\,F=\{x\in\mathbb{R}^{k}\;:\;F(x)\not=\emptyset\}$ and

If ${\rm gph}\,F$ is a convex set in $\mathbb{R}^{k}\times\mathbb{R}^{\ell}$, then $F$ is said to be a convex multifunction. In that case, the coderivative of $F$ at $(\bar{x},\bar{y})\in{\rm{gph}}\,F$ is the multifunction $D^{*}F(\bar{x},\bar{y}):\mathbb{R}^{\ell}\rightrightarrows\mathbb{R}^{k}$ with

for all $v\in\mathbb{R}^{\ell}$ (see, e.g., (Mordukhovich_Book_I, , Section 1.2) and An_Yen_2015 ).

From now on, let $K$ be a closed convex cone in $\mathbb{R}^{m}$. For any $y^{1},y^{2}$ from $\mathbb{R}^{m}$, one writes $y^{1}\geq_{K}y^{2}$ if $y^{1}-y^{2}\in K$. Similarly, one writes $y^{1}>_{K}y^{2}$ if $y^{1}-y^{2}\in{\rm int}K$, where ${\rm int}K$ denotes the interior of $K$.

Given a matrix $A\in\mathbb{R}^{m\times n}$, vectors $b\in\mathbb{R}^{m}$ and $c\in\mathbb{R}^{n}$, we consider the primal conic linear optimization problem

Following Bental and Nemirovski (BN_2001, , p. 52), we call

the dual problem of (P). The feasible region, and the solution set, and the optimal value of (P) are denoted respectively by $\mathcal{F}{\rm(P)}$, $\mathcal{S}{\rm(P)}$, and $v{\rm(P)}$. The feasible region, the solution set, and the optimal value of (D) are denoted respectively by $\mathcal{F}{\rm(D)}$, $\mathcal{S}{\rm(D)}$, and $v({\rm D})$. By definitions, one has

and $v({\rm D})=\sup\{b^{T}y\;:\;y\in\mathbb{R}^{m},\,A^{T}y=c,\,y\geq_{K^{*}}0\}.$ By a standard convention, $\inf\emptyset=+\infty$ and $\sup\emptyset=-\infty$.

Thanks to the weak duality theorem in conic linear programming (BN_2001, , Proposition 2.3.1), one has $c^{T}x\geq b^{T}y$ for any $x\in\mathcal{F}{\rm(P)}$ and $y\in\mathcal{F}{\rm(D)}$. Hence, $v({\rm P})\geq v({\rm D})$. By constructing suitable examples, we can show that the strict inequality $v({\rm P})>v({\rm D})$ is possible, i.e., a conic linear program may have a duality gap. Thus, the equality $v({\rm P})=v({\rm D})$ is guaranteed only if one imposes a certain regularity condition. In what follows, we will rely on the regularity condition called strict feasibility, which was intensively employed by Bental and Nemirovski BN_2001 .

The main assertion of the strong duality theorem in conic linear programming can be stated as follows.

The following analogues of the Farkas lemma (Rockafellar_1970, , p. 200) will be used repeatedly in the sequel.

Proof To prove assertion (a), suppose that $Ax\geq_{K}b$ and there exists a vector $y\in K^{*}$ satisfying $A^{T}y=c$ and $b^{T}y\geq\alpha$. Then we have

where the last inequality in (1) is valid because $Ax\geq_{K}b$ and $y>_{K^{*}}0$. Combining this with the assumption $b^{T}y\geq\alpha$, yields $c^{T}x\geq\alpha$. Conversely, suppose that $c^{T}x\geq\alpha$ for all $x$ satisfying $Ax\geq_{K}b$. It is clear that (P) is strictly feasible and $v({\rm P})\geq\alpha$. By the first assertion of Lemma 1, (D) has a solution and one has $v({\rm P})=v({\rm D})$. Let $y$ be a solution of (D). Then we have $y\in K^{*}$, $A^{T}y=c$ and $b^{T}y=v({\rm D})=v({\rm P})\geq\alpha$. So, assertion (a) holds true.

Now, to prove assertion (b), suppose that $A^{T}y=c,\,y\geq_{K^{*}}0$ and there exists a vector $x\in\mathbb{R}^{n}$ satisfying $Ax\geq_{K}b$ and $c^{T}x\leq\alpha$. Then, we have

where the last inequality in (2) is valid because $Ax\geq_{K}b$ and $y\geq_{K^{*}}0$. Since $c^{T}x\leq\alpha$, this yields $b^{T}y\leq\alpha$. Conversely, suppose that $b^{T}y\leq\alpha$ for all $y$ satisfying $A^{T}y=c,\,y\geq_{K^{*}}0$. So, (D) is strictly feasible by our assumption and $v({\rm D})\leq\alpha$. By the second assertion of Lemma 1, (P) has a solution and $v({\rm P})=v({\rm D})$. Let $x$ be a solution of (P). Then we have $Ax\geq_{K}b$ and $c^{T}x=v({\rm P})=v({\rm D})\leq\alpha$. This justifies the validity of (b). $\hfill\Box$

The optimal value $v({\rm P})$ depends on the parameters $A$, $b$, and $c$. Following Gauvin Gauvin_2001 , who studied differential stability properties of linear programming problems, we consider the functions $\varphi(b):=\inf\{c^{T}x\;:\;x\in\mathbb{R}^{n},\,Ax\geq_{K}b\}$ and $\psi(c):=\inf\{c^{T}x\;:\;x\in\mathbb{R}^{n},\,Ax\geq_{K}b\}.$ Note that $\varphi(\cdot)$ is the optimal value function of (P) under right-hand-side perturbations of the constraint set and $\psi(\cdot)$ is the optimal value function of (P) under perturbations of the objective function.

According to a remark given in the paper of An and Yen (An_Yen_2015, , p. 113), we know that $\varphi(\cdot)$ is a convex function. The next proposition presents additional continuity properties of $\varphi$.

Proof  On one hand, since (P) is strictly feasible, one selects a point $x^{0}\in\mathbb{R}^{n}$ satisfying $Ax^{0}>_{K}b$, i.e., $Ax^{0}-b\in{\rm int}K$. Then there exists a convex open neighborhood of $b$ satisfying $Ax^{0}-b^{\prime}\in{\rm int}K$ for every $b^{\prime}\in V$. So, $x^{0}$ is a feasible point of the problem

for all $b^{\prime}\in V$. Moreover $\varphi(b^{\prime})\leq c^{T}x^{0}<+\infty$. On the other hand, applying the weak duality theorem (BN_2001, , Proposition 2.3.1) for the problems (${\rm P}_{b^{\prime}}$) and

one has $c^{T}x\geq(b^{\prime})^{T}y$ for all $x\in\mathbb{R}^{n}$ satisfying $Ax\geq_{K}b^{\prime}$ and for every $y\in\mathbb{R}^{m}$ satisfying $A^{T}y=c$, $y\geq_{K^{*}}0$. Therefore,

This means that

So, we have $\varphi(b^{\prime})>-\infty$ because

where the last inequality in (3) is valid as $\mathcal{S}({\rm D})$ is nonempty (see Lemma 1). It follows that $\varphi(b^{\prime})$ is finite for every $b^{\prime}\in V$, i.e., $V\subset{\rm int}({\rm dom}\varphi)$. Combining this with the convexity of $\varphi$, by (Rockafellar_1970, , Theorem 24.7) we can asserts that $\varphi$ is locally Lipschitz on $V$. $\hfill\Box$

First, let us show that the solution set of (D) possesses a remarkable stability property, provided that the assumptions of Theorem 3.1 are satisfied.

Proof Since (P) is strictly feasible, there is $x^{0}\in\mathbb{R}^{n}$ satisfying $Ax^{0}-b\in{\rm int}K$. Therefore, there is a real number $\varepsilon>0$ satisfying $Ax^{0}-b^{\prime}\in{\rm int}K$ for all $b^{\prime}\in B(b,\varepsilon)$. According to Theorem 3.1, without loss of generality we can assume that $\varphi(b^{\prime})$ is finite for all $b^{\prime}\in B(b,\varepsilon)$. Fix a vector $b^{\prime}\in B(b,\varepsilon)$. Then, the problem (${\rm P}_{b^{\prime}}$) is strictly feasible and $v\eqref{P_b'}>-\infty$. So, applied for the pair problems (${\rm P}_{b^{\prime}}$) and (${\rm D}_{b^{\prime}}$) Lemma 1 asserts that $\mathcal{S}\eqref{D_b'}$ is nonempty and

Since $\mathcal{F}\eqref{D_b'}=\mathcal{F}({\rm D})$, this implies $\mathcal{S}\eqref{D_b'}=\big{\{}y\in\mathcal{F}({\rm D})\;:\;(b^{\prime})^{T}y=\varphi(b^{\prime})\big{\}}.$ Clearly, $\mathcal{S}\eqref{D_b'}$ is a closed set. To prove that $\mathcal{S}\eqref{D_b'}$ is compact by contradiction, let us suppose that $\mathcal{S}\eqref{D_b'}$ is unbounded. Select a sequence $\{y^{k}\}$ in $\mathcal{S}\eqref{D_b'}$ satisfying the condition $\lim\limits_{k\to\infty}\|y^{k}\|=+\infty$. Define $\widetilde{y}^{k}=\|y^{k}\|^{-1}y^{k}$. Since $\|\widetilde{y}^{k}\|=1$, without loss of generality we can assume that the sequence $\{\widetilde{y}^{k}\}$ converges to a vector $\widetilde{y}$ with $\|\widetilde{y}\|=1$. Since the sequence $\{y^{k}\}$ is contained in the closed cone $K^{*}$, one has $\widetilde{y}\in K^{*}$. Observe that

and

It follows that $(Ax^{0}-b^{\prime})^{T}\widetilde{y}={(x^{0})}^{T}A^{T}\widetilde{y}-(b^{\prime})^{T}\widetilde{y}=0$. Meanwhile, since $Ax^{0}-b^{\prime}\in{\rm int}K$ and $\widetilde{y}\in K^{*}\setminus\{0\}$, $(Ax^{0}-b^{\prime})^{T}\widetilde{y}>0$. We have obtained a contradiction. Thus, $\mathcal{S}\eqref{D_b'}$ is bounded; hence it is compact. $\hfill\Box$

The following question arises naturally from Proposition 1: If (D) is strictly feasible and $v(\rm D)<+\infty$, then the solution set $\mathcal{S}({\rm P})$ is nonempty and compact, or not? By the second assertion of Lemma 1, if (D) is strictly feasible and $v(\rm D)<+\infty$, then $\mathcal{S}({\rm P})$ is nonempty. However, $\mathcal{S}({\rm P})$ may not be a compact set. The next example is an illustration for this statement.

The following result gives a formula for the directional derivative of $\varphi$ at $b$ in a direction $d$.

Proof Fix a direction $d\in\mathbb{R}^{m}$, consider the function $g:\mathbb{R}\to\overline{\mathbb{R}}$ given by

where $t\in\mathbb{R}$. Let $f(x,t)=c^{T}x$ for all $(x,t)\in\mathbb{R}^{n}\times\mathbb{R}$. Clearly, $f$ is convex and continuous on $\mathbb{R}^{n}\times\mathbb{R}$. Define the multifunction $G:\mathbb{R}\rightrightarrows\mathbb{R}^{n}$ by setting

Note that $G$ is a convex multifunction. Consider the parametric convex optimization problem $\min\{f(x,t)\;:\;x\in G(t)\},$ which depends on the parameter $t\in\mathbb{R}$, and observe that $g(\cdot)$ is the optimal value function of this problem. By a remark given in (An_Yen_2015, , p. 113) we know that $g(\cdot)$ is a convex function. Applying a result of An and Yen (An_Yen_2015, , Theorem 4.2) for $\bar{t}:=0$ and for a solution $\bar{x}$ of $({\rm P})$, we have

Since $\partial f(x,t)=\left\{\begin{bmatrix}c\\ 0\end{bmatrix}\right\}$, this equality yields

By the definition of coderivative for convex multifunctions,

Let

Clearly, the inequality $Ax-td\geq_{K}b$ is equivalent to $\widetilde{A}\widetilde{x}\geq_{K}b$. So, from (6) one gets

Since (P) is strictly feasible, there exists $x^{0}\in\mathbb{R}^{n}$ satisfying $Ax^{0}>_{K}b$. Setting $\widetilde{x}^{0}=\begin{bmatrix}x^{0}\\ 0\end{bmatrix}$, we have $\widetilde{A}\widetilde{x}^{0}>_{K}b$. Then, applying the Farkas-type result in Lemma 2(a) to the inequality system $\widetilde{A}\widetilde{x}\geq_{K}b$ and the vector $\widetilde{c}$, we can assert that $\widetilde{c}^{T}\widetilde{x}\geq\alpha$ for all $\widetilde{x}\in\mathbb{R}^{n}\times\mathbb{R}$ such that $\widetilde{A}\widetilde{x}\geq_{K}b$ iff there exists a vector $y\in K^{*}$ satisfying $\widetilde{A}^{T}y=\widetilde{c}$ and $b^{T}y\geq\alpha$. Since

the equality $\widetilde{A}^{T}y=\widetilde{c}$ is equivalent to the conditions $A^{T}y=c$ and $d^{T}y=t^{*}$. Therefore, combining (7) with the description of the feasible region of the dual problem (D), we have

Since (P) is strictly feasible and (P) has a solution, by the strong duality theorem in Lemma 1, (D) has a solution and $v({\rm D})=v({\rm P})$. Hence, by (8),

Combining (9) with (5) gives $\partial g(0)=\{d^{T}y\;:\;y\in\mathcal{S}({\rm D})\}.$ Then, by the well-known result on the relationships between the directional derivative of a convex function (Rockafellar_1970, , Theorem 23.4), we obtain

where the last equality in (10) is valid because $\mathcal{S}({\rm D})$ is compact. Finally, using (10) and the fact that $\varphi^{\prime}(b;d)=g^{\prime}(0;1)$, we get the desired formula (4). $\hfill\Box$

Based on Theorem 4.1, the next proposition provides us with a formula for the subdifferential of $\varphi$ at $b$.

Proof Take a vector $p\in\partial\varphi(b)$. For any $d\in\mathbb{R}^{m}$ and $t>0$, from the inequality $\varphi(b+td)-\varphi(b)\geq p^{T}(td)$ it follows that $\dfrac{1}{t}\left(\varphi(b+td)-\varphi(b)\right)\geq p^{T}d.$ Letting $t\to 0^{+}$, one has $\varphi^{\prime}(b;d)\geq p^{T}d.$ Combining this with the equation (4) gives

To show that $p$ belongs to $\mathcal{S}({\rm D})$, suppose the contrary: $p\notin\mathcal{S}({\rm D})$. By $\mathcal{S}({\rm D})$ is a nonempty convex set, by the strongly separation theorem (see, e.g., (Rockafellar_1970, , Corollary 11.4.2)), there exists a vector $d\in\mathbb{R}^{m}$ such that $\max\limits_{y\in\mathcal{S}({\rm D})}y^{T}d<p^{T}d$. This contradicts with (11). We have thus proved that $\partial\varphi(b)\subset\mathcal{S}({\rm D})$. To obtain the opposite inclusion, take any $p\in\mathcal{S}({\rm D})$. For all $d\in\mathbb{R}^{m}$, by (4),

Besides, by (Rockafellar_1970, , Theorem 23.1) one gets

Combining (12) with (13) we have $\varphi(b+d)-\varphi(b)\geq p^{T}d$ for all $p\in\mathbb{R}^{m}$. So, $p\in\partial\varphi(b)$. The proof is complete. $\hfill\Box$

Roughly speaking, the following statement gives an analogue of (Gauvin_2001, , Theorem 1) for conic linear programs.

Proof By Lemma 1, (D) has a solution and $v({\rm D})=v({\rm P})$; hence $v({\rm D})=\varphi(b)$. On one hand, applying the weak duality theorem (BN_2001, , Proposition 2.3.1) for the problems

and

one has $c^{T}x\geq(b+td)^{T}y$ for all $x\in\mathbb{R}^{n}$ satisfying $Ax\geq_{K}b+td$ and for every $y\in\mathbb{R}^{m}$ satisfying $A^{T}y=c$, $y\geq_{K^{*}}0$. Consequently,

On the other hand, it is clear that

Combining (15) and (16), one gets