Lagrangian Duality in Convex Conic Programming with Simple Proofs

Abstract

In this paper, we study Lagrangian duality aspects in convex conic programming over general convex cones. It is known that the duality in convex optimization is linked with specific theorems of alternatives. We formulate and prove the strong alternative theorems to the strict feasibility and analyze the relation between the boundedness of the optimal solution sets and the existence of the relative interior points in the feasible set. We also provide sufficient conditions under which the duality gap is zero and the optimal solution sets are unbounded. As a consequence, we obtain several new sufficient conditions that guarantee the strong duality between primal and dual convex conic programs. Our proofs are based only on fundamental convex analysis and linear algebra results.

1 Introduction

Duality in convex conic programming has historically been studied in a general setting, starting with [1], and followed by other authors (see [2] for more references). In [3] and [4], the authors use minimal cone to transform the original problem to an equivalent one, for which the Slater constraint qualification (SCQ) holds and hence the strong duality property is satisfied. However, as mentioned in [5], the facial reduction procedure to obtain a minimal cone is computationally unsatisfactory.

Duality theory in convex conic programming, with the focus on specific subclasses, has been revisited by many authors since the IPM revolution in 1984. One of such papers is [6], where duality results for linear programming are obtained from the perspective of the IPM methodology. In particular, it is shown that the primal (dual) feasibility, together with the dual (primal) strict feasibility, is equivalent to the nonemptiness and boundedness of the primal (dual) optimal solution set, respectively (Theorem 3.2, [6]). Duality theory for semidefinite programming (SDP) is studied in connection with IPM in [2]; see also [7] and [8] for a survey. Simple proofs for the extension of the result of Theorem 3.2 in [6] to SDP are given in [9].

The papers mentioned in the paragraph above study the Lagrangian dual (considered in the convex optimization textbooks, such as [10, 11], or [12]) that requires SCQ for the strong duality to hold. Failure of strong duality motivated other authors, who attempted to construct a primal-dual pair satisfying strong duality without any constraint qualification. The extended Lagrange-Slater dual was proposed in [5] for semidefinite programming. In the paper [13], the facial reduction procedure of [4] was applied to obtain strong duality for convex conic problems over symmetric cones. Paper [14] dealt with general convex conic programs, and it was shown that the minimal representation of the problem guarantees SCQ and therefore also strong duality. The same approach was applied in [15] for the class of copositive programming problems.

Contrary to the approaches mentioned in the paragraph above, we study the standard Lagrangian primal-dual pair of convex conic programming problems (see [2, 10,11,12]), where strong duality may fail.

The major results of our paper may be summarized as follows. We formulate and prove new strong theorems of alternatives, which give equivalent conditions to the strict feasibility of the primal and dual problem (Theorems 8 and 13). We show that the boundedness of the (non-empty) optimal solution is equivalent to the existence of a relative interior point in the set of feasible solutions of the dual counterpart (Theorem 21). As a consequence, we obtain new sufficient conditions for strong duality. We also derive different necessary and sufficient conditions for strong duality which guarantee that the particular set of optimal solutions is nonempty but unbounded (Theorem 26). In the proofs, we only use basic results from linear algebra and convex analysis, which might be useful to practitioners. This paper was motivated by the following.

Firstly, analogous duality results shown for semidefinite programs in, e.g., [8] and [9] appeared to be useful in polynomial optimization and Lasserre’s hierarchies [16], where especially the boundedness of the (nonempty) optimal solutions set appeared to be a useful sufficient condition for strong duality. We offer a generalization of the results to the case of conic problems over convex cones with no specific structure.

Secondly, every standard convex programming problem has a conic reformulation (like the one considered in our paper). Even if standard SCQ (and strong duality) fails for the classic formulation, it may hold in the conic reformulation (see Example 20 in Sect. 4), which justifies the conic modeling approach. New insights offered in our paper might be helpful for analyzing other subclasses of convex conic problems.

This paper is organized as follows. Section 2 is the preliminary section, which includes the basic notation, the review of some known useful properties, and several results related to the closedness of the linear image of the convex cone as well as the Minkowski sum of a convex cone and a linear subspace. In Sect. 3, we present four theorems of alternatives. Two of them are the known generalized Farkas type theorems that require a closedness assumption (discussed in Sect. 3) to hold in the strong version. In addition to these, we present two new ones that give equivalent conditions to the existence of the relative interior point in the primal (dual) feasible set. The theorems of alternatives served as a tool for deriving the strong duality results, included in Sect. 4. We summarize the known duality results and relate them to our findings, which include new sufficient conditions for strong duality and properties of the optimal solution sets. Section 5 concludes.

2 Preliminaries

2.1 Properties of Cones and Dual Cones

A subset K of a finite-dimensional vector space $\mathbb {R}^n$ is called a cone if $\forall x\in K$ and $\forall \alpha \ge 0$ it holds $\alpha x\in K$. A convex cone is closed under vector addition, that is, $\forall x, y\in K$, we have $x+y\in K$. Sometimes, additional properties can also be imposed: a cone is called pointed if it does not contain a straight line; it is called solid, if its interior is nonempty. A convex, closed, pointed, and solid cone is called a proper cone (see, e.g., [11] and [12]). Denote $lin(K):=K+(-K)$ the smallest linear subspace containing the cone K, and ${sub}(K):=K\cap (-K)$, the largest linear subspace contained in K. It can be seen that a convex cone is pointed if and only if $sub(K)=\{0\}$, and it is solid if and only if $lin(K)=\mathbb {R}^n$.

For a nonempty set $C \subseteq \mathbb {R}^n$, the recession cone $R_C$ of the set C is defined as $R_C:= \{ d \ | \ x + \gamma d \in C, \forall x \in C, \ \forall \lambda \ge 0\}$. Vectors included in $R_C$ are called directions of recession of the set C. The recession cone of a nonempty convex set is a convex cone; moreover, if the convex set is closed, its recession cone is closed. Clearly, if C is nonempty and bounded, then $R_C = \{ 0 \}$; however, the converse may not hold true unless C is closed (see, e.g., Proposition 1.5.1 in [17]).

The dual cone of a cone K¹ is the set $K^*=\{ y \ |\ x^\top y\ge 0, \ \forall x \in K \}$. Its fundamental properties can be found, e.g., in [11, 12, 18].

A cone K will be called trivial if it is a linear subspace, i.e., $K=sub(K)=lin(K)$; otherwise, it will be called nontrivial. If a cone K is trivial, then clearly $K^*=K^{\bot }$. An important tool in conic duality theory is the bipolar theorem (see, e.g., [18, Theorem 14.1]; [19, Proposition 4.2.6]) and its consequences.

Theorem 1

(Bipolar theorem)

If K is a convex cone, then $K^{**}=cl(K)$.

Using the characterization of lin(K) and sub(K) and the bipolar theorem, it can be easily shown that the linear subspaces are linked in the following way (see[20, Corollary 1]):

$$\begin{aligned} {sub}(K^*)&=\{ y\in K^*, \ | \ x^{\top }y=0, \ \forall x\in K\}=lin(K)^{\bot };\end{aligned}$$

(1)

$$\begin{aligned} {sub}(cl(K))&= \{ z\in cl(K), \ | \ z^{\top }y=0, \ \forall y\in K^*\}=lin(K^*)^{\bot }. \end{aligned}$$

(2)

Note that when (2) is applied to $K^*$ and combined with (1) and the bipolar theorem, it follows that $lin(K) = lin(cl(K))$.

In the following, we list a few simple properties of $lin(\cdot )$ and $sub(\cdot )$ of a convex cone intersected with a linear subspace:

$$\begin{aligned} sub(V \cap K)&=V \cap sub (K), \end{aligned}$$

(3)

$$\begin{aligned} V \cap [K \setminus sub(K)]&=(V \cap K)\setminus sub(V\cap K), \end{aligned}$$

(4)

In this paper, we will deal with the primal-dual pair of convex conic programs, where the cone K satisfies the following:

Assumption 1

The cone K is a nontrivial convex cone.

Clearly, for nontrivial convex cones, the following equivalent conditions hold: $lin(K)\setminus K\ne \emptyset$, $K\setminus sub(K)\ne \emptyset$.

2.2 Relative Interior of a Convex Cone

For a general convex cone K, the relative interior relint(K) is defined as the interior of K with respect to the subspace topology on lin(K). The convexity property allows for the following definition:

$$\begin{aligned} {relint}(K)=\{ x\in K \ | \ \forall v\in lin(K)\ \exists \lambda >0: x+\lambda v\in K \}. \end{aligned}$$

Another characterization of the relative interior of K was introduced in [20, Theo-rem 2]:

$$\begin{aligned} relint(K) = \{ x \in K \ | \ x^{\top } y > 0, \ \forall y \in K^* \setminus sub(K^*) \}. \end{aligned}$$

(5)

From characterization (5) and the bipolar theorem, we obtain a characterization of the relative interior of the dual cone $K^*$:

$$\begin{aligned} relint(K^*)=\{ y\in K^* \ | \ x^{\top }y>0, \ \forall x\in cl(K) \setminus sub(cl(K)) \}. \end{aligned}$$

(6)

Since $0\in K$, from the definition of the dual cone and characterization (5), itfollows that

$$\begin{aligned} relint(K) = K + relint(K) = cl(K) + relint(cl(K)). \end{aligned}$$

(7)

Now, we recall a few known properties (see [17, 18], and [21]). Assume that $K_1$ and $K_2$ are convex cones. Then, for any $K_1, K_2\subseteq \mathbb {R}^n$, it holds $relint (K_1+K_2)=relint(K_1)+ relint(K_2)$; moreover, if $relint(K_1) \cap relint(K_2) \ne \emptyset$, then $relint(K_1 \cap K_2) = relint(K_1) \cap relint(K_2)$, and for any $K_1\subseteq \mathbb {R}^m, K_2\subseteq \mathbb {R}^n$, it holds $relint (K_1\times K_2)=relint(K_1)\times relint(K_2)$. Finally, assume that K is a convex cone and A is an $m \times n$ matrix, then $A(relint(K)) = relint(A(K))$.²

2.3 Primal and Dual Convex Conic Programs

Given vectors $c\in \mathbb {R}^n$, $b\in \mathbb {R}^m$, an $m\times n$ matrix A and a convex cone $K\subset \mathbb {R}^n$, the convex conic programming problem in standard form is formulated as

$$\begin{aligned} \begin{array}{rl} \min &{} c^{\top }x \\ \mathrm {s.t.} &{} Ax=b \\ &{} x\in {K}. \end{array} \end{aligned}$$

(8)

The set of primal feasible points and the set of primal strictly feasible points are denoted by $\mathcal {P}=\{ x\in {K} \ | \ Ax=b \}$ and $\mathcal {P}^0=\{ x\in \hbox {relint} (K) \ | \ Ax=b \}$, respectively. Furthermore, we define the optimal value of the problem (8) as $p^*=\inf \{c^{\top }x \ | \ x\in \mathcal {P}\}$ if $\mathcal {P}\ne \emptyset$ and $p^*=+\infty$ otherwise. The primal optimal solution set is then $\mathcal {P}^*=\{ x\in \mathcal {P} \ | \ c^{\top }x=p^* \}.$ Using the concept of Lagrangian duality and the standard techniques, one can derive the dual of problem of (8):

$$\begin{aligned} \begin{array}{rl} \max &{} b^{\top }y \\ \hbox {s.t.} &{} A^{\top }y+s=c \\ &{} s\in K^*. \end{array} \end{aligned}$$

(9)

The set of all dual feasible points of (9) is $\mathcal {D}=\{ (y,s)\in \mathbb {R}^m\times K^* \ | \ A^{\top }y+s=c \}$, and the set of all dual strictly feasible points is $\mathcal {D}^0=\{ (y,s) \in \mathbb {R}^m\times \hbox {relint} (K^*) \ | \ A^{\top }y+s=c \}$. If $\hbox {rank}(A)=m$, then there is one-to-one correspondence between the dual variables y and s, that is, if $(y_1, s), (y_2,s)\in \mathcal {D}$, then $y_1=y_2$.³ The optimal value of the problem (9) is defined as $d^*=\sup \{b^{\top }y \ | \ (y,s)\in \mathcal {D}\}$ if $\mathcal {D}\ne \emptyset$ and $d^*=-\infty$ otherwise. Finally, the dual optimal solution set is denoted by $\mathcal {D}^*$, i.e., $\mathcal {D}^*=\{ (y,s) \in \mathcal {D} \ | \ b^{\top }y=d^* \}.$ The weak duality property follows directly from the definition of the problems and the dual cone: for each $x\in \mathcal {P}$ and $(y,s)\in \mathcal {D}$, it holds $x^{\top }s=c^{\top }x-b^{\top }y\ge 0.$

In the following chapters, we will be working with recession cones of $\mathcal {P}$, $\tilde{\mathcal {D}} =\{ s \ | \ (y,s)\in \mathcal {D}\}$, $\mathcal {P}^*$ and $\mathcal {\tilde{D}}^* = \{ s^* \ | \ (y^*,s^*) \in \mathcal {D}^* \}$. It can be easily verified that the recession cones of sets $\mathcal {P}$ and $\tilde{\mathcal {D}}$, supposing that these sets are nonempty, are

$$\begin{aligned} R_{\mathcal {P}}&= \{ d \ | \ Ad = 0, \ d \in K \} = \mathcal {N}(A) \cap K, \end{aligned}$$

(10)

$$\begin{aligned} R_{\mathcal {\tilde{D}}}&= \{ v \ | \ v \in \mathcal {S}(A^\top ), \ v \in K^* \} = \mathcal {S}(A^\top ) \cap K^*, \end{aligned}$$

(11)

Now, we define the extended matrices $\textbf{A}_c=(A^{\top }\ c)^{\top } \in \mathbb {R}^{(m+1) \times n}$ and $\textbf{A}_b=(A-b) \in \mathbb {R}^{m \times (n+1)}$. The recession cones of sets $\mathcal {P}^*$ and $\mathcal {\tilde{D}}^* = \{ s^* \ | \ (y^*,s^*) \in \mathcal {D}^* \}$, supposing that these sets are nonempty, are

$$\begin{aligned} R_{\mathcal {P^*}}&= \{ d \ | \ Ad = 0, \ c^{\top } d = 0, \ d \in K \} = \mathcal {N}(\textbf{A}_c) \cap K, \end{aligned}$$

(12)

$$\begin{aligned} R_{\mathcal {\tilde{D}^*}}&= \{ v \ | \ (v^\top ,0)^\top \in \mathcal {S}(\textbf{A}_b^\top ) \cap (K^{*} \times \{ 0\}) \}. \end{aligned}$$

(13)

2.4 Closedness of the Linear Image of a Convex Cone

Linear programs, i.e., conic linear programs for which the cone K is polyhedral, are characterized by “ideal” duality theory. This is closely related to the famous Farkas theorem of alternatives [22] and the fact that convex polyhedral cones are finitely generated and hence their linear images form closed cones. This guarantees that the alternatives appearing in Farkas’ lemma are strong, i.e., one and only one of the alternatives holds. However, in the generalized versions of the Farkas lemma, the alternatives are weak (i.e., at most, one of the two holds), and the closedness of the linear image of the related convex cone becomes an additional assumption.

In this section, we summarize the sufficient conditions for the closedness of the linear image of a convex cone. We start with the following lemma.⁴

Table 1 Equivalent conditions of Lemma 2 formulated for specific linear subspaces and cones appearing in the primal and dual conic linear programs (8) and (9). Conditions (i-c)–(iii-c) correspond to the special case of cl(K) being pointed, and conditions (i-d)–(iii-d) correspond to the special case of K being solid

Lemma 2

Let $L\subseteq \mathbb {R}^n$ be a linear subspace, and let $K\subset \mathbb {R}^n$ be a cone satisfying Assumption 1. Then, the following statements are equivalent:

• (i) $L+K=L+{lin}(K)$

• (ii) $L\cap {relint}(K)\ne \emptyset$

• (iii) $L^{\bot } \cap [K^*\setminus {sub}(K^*)]=\emptyset$

Note that if K is solid, then the statements (i), (ii),  and (iii) can be simplified to $L+K=\mathbb {R}^n$, $L\cap int(K)\ne \emptyset$, $L^{\bot }\cap K^*=\{0\}$. The paper [23] briefly discusses the appearance of the equivalent conditions in Lemma 2 in literature, expressed in terms of $\mathcal {N}(A)$, or $\mathcal {S}(A^{\top })$, i.e., L corresponding to the null space or the range of the $m\times n$ matrix A. For the reader’s convenience, we formulate the alternative expressions of the equivalent conditions $(i)-(iii)$ of Lemma 2 in Table 1.

Remark 3

It can be easily seen that (i-a)–(iii-a) and (i-b)–(iii-b) (similarly (i-c)–(iii-d) and (i-d)–(iii-d)) are weak alternatives: at most, one of them holds. Clearly, if (i-a)–(iii-a) (or (i-c)–(iii-c)) holds, then (i-b)–(iii-b)) (or (i-d)–(iii-d)) does not hold. However, they are not strong alternatives, as demonstrated in the following example: let $A=(1 \ 0 \ 1)$ and

$$\begin{aligned} K=K^*=cl(K) := \{ (x_1,x_2,x_3)^{\top } \ | \ \sqrt{x_1^2+x_2^3} \le x_3 \}. \end{aligned}$$

$\mathcal {N}(A)$ is generated by $(-1,0,1)^{\top }, (0,1,0)^{\top }$, and hence, neither (iii-c) nor(iii-d) holds.

Remark 4

Conditions (i-a), (iii-a), (i-b), and (iii-b) can be formulated in terms of recession cones of $\mathcal {P}$ and $\mathcal {\tilde{D}}$ (see (10) and (11)), provided that these sets are nonempty, as follows: condition (i-a) is equivalent to $R_{{\mathcal {P}}}^* = cl(S(A^T)+K^*)$ being a linear subspace and condition (iii-a), under a condition of closedness of K, is equivalent to $R_\mathcal {P}$ being a linear subspace. Condition (i-b), with requirement that K is closed, is equivalent to $R_{\mathcal {\tilde{D}}}^* = cl(\mathcal {N}(A) + K)$ being a linear subspace and condition (iii-b) is equivalent to $R_{\mathcal {\tilde{D}}}$ being a linear subspace.

Table 1 lists conditions under which a linear image of a convex cone is closed: it was shown in [18, Theorem 9.1] that (iii-a) implies $cl(A(K))=A(cl (K))$. On the other hand, since $A(\mathcal {N}(A))=\{0\}$, it can be easily seen that (i-b) implies

$$\begin{aligned} A(K)=A(\mathcal {N}(A)+K)=A(\mathcal {N}(A)+lin(K))=A(lin (K)), \end{aligned}$$

and hence, in this case, A(K) is also closed (it is a linear subspace).

Moreover, a known result, often referred to as Theorem of Abrams, states that for a nonempty set $S\subseteq \mathbb {R}^n$ and a linear map given by matrix A, it holds

$$\begin{aligned} A(S) \; \text {is closed}\; \Leftrightarrow \mathcal {N}(A) +S \; \text {is closed}; \end{aligned}$$

see [24, Proposition 3.1].

We can summarize the results in the following theorem:

Theorem 5

Assume that K satisfies Assumption 1.

• (a) If any of the conditions (i-a), (ii-a), (iii-a) holds, then A(cl(K)) is closed and $\mathcal {S}(A^{\top })+K^*$ is a linear subspace.

• (b) If any of the conditions (i-b), (ii-b), and (iii-b) holds, then $A(cl(K))=A(K)$ is a linear subspace and $\mathcal {S}(A^{\top })+K^*$ is closed.

Remark 6

Consider the second-order cone K and A from Remark 3. It can be easily seen that in this case,

$$\begin{aligned} A(K)=\{ u+w \ | \ (u,v,w)^{\top } \in K\}=\mathbb {R}_+, \end{aligned}$$

and hence, it is closed. This shows that the conditions in Table 1 are not necessary.

For more results and references, we refer the reader to [23], where the sufficient conditions for the closedness of a linear image of a convex cone and the Minkowski sum were studied in a more general setting, and the conditions were shown to be also necessary for a special class of cones.

3 Theorems of Alternatives

In this section, we present four theorems of alternatives for linear systems over cones. They are divided into two groups, depending on whether, regarding the (strict) feasibility, they are related to the primal or the dual conic program. Two of them are known as the Farkas lemma, and the alternatives presented in the theorems are weak in general. For strong alternatives, an additional assumption is required. Note that in the Farkas lemma, one alternative is exactly the feasibility of the primal (dual) convex conic program. We also formulate and prove a new different (primal-dual) pair of theo-rems of alternatives, where one alternative is the strict feasibility of the primal (dual) convex conic program. The alternatives in these theorems are strong (no additional assumption is required).

The first theorem is a generalization of the famous Farkas lemma for linear systems [22]. Various forms of the theorem have been studied within the last decades, also with the connection to linear matrix inequalities and semidefinite programming; see [25] and [2]. For general conic programs, it was formulated by many authors in various forms; see, e.g., [26, 27], or [28] in more general terms.

Theorem 7

(Generalized Farkas lemma)

Assume that $K\subseteq \mathbb {R}^n$ is a cone satisfying Assumption 1, A is a given $m\times n$, ($m\le n$) matrix, and $b\in \mathbb {R}^m$ and $c\in \mathbb {R}^n$ are given vectors. At most, one of the following statements is true:

• (I) $\exists x\in K: {A}x= b$;

• (II) $\exists z: {A}^{\top }z\in K^*$ and $z^{\top }b< 0$.

Moreover, if the convex cone A(cl(K)) (or alternatively the Minkowski sum $cl(K)+\mathcal {N}(A)$) is closed, then exactly one of the statements is true.

In the following, we establish and prove a new theorem of alternatives, which deals with the relative interior of the cone. It provides a strong alternative (and therefore also an equivalent condition) to the strict feasibility of the primal program (8).

Theorem 8

Assume that $K\subseteq \mathbb {R}^n$ is a cone satisfying Assumption 1, A is a given $m\times n$, ($m\le n$) matrix, and $b\in \mathbb {R}^m$ and $c\in \mathbb {R}^n$ are given vectors. Exactly one of the following statements is true:

• (I) $\exists x\in relint(K): {A} x= b$;

• (II) $\left[ \exists z: A^{\top } z \in K^{*} {\setminus } sub(K^{*}) \ and \ z^{\top } b \le 0 \right] \$ or $\left[ \exists z: A^{\top } z \in sub(K^{*}) \ and \ z^{\top } b \ne 0.\right]$

Proof

First, we will show that I and II cannot hold at once. Assume the opposite, then $\bar{z}^\top A \bar{x} \le 0$ for some $\bar{x}$ and $\bar{z}$ that fulfill I and II, respectively. However, from the characterization (5) and (1), we obtain $\bar{z}^\top A \bar{x} >0$, which is a contradiction. Thus, I implies $\lnot II$.

Now, we will show that $\lnot I$ implies II. Suppose that I does not hold, or equivalently $b\notin A(relint(K))$. With respect to vector b, there are two cases to consider:

• 1. $b \in lin(A(K)) \setminus A(relint(K))$

• 2. $b \notin lin(A(K))$

Case 1 implies $lin(A(K)) \setminus A(relint(K)) \ne \emptyset$, and hence, $A(relint(K))= relint(A(K))$ is nontrivial. This implies that cl(A(K)) is nontrivial and so is the dual cone $[A(K)]^* = \{ z \ | \ A^\top z \in K^* \}$, thus $[A(K)]^* {\setminus } sub([A(K)]^*)= \{ z \ | \ A^\top z \in K^* {\setminus } sub(K^*) \} \ne \emptyset$. Now, if $b \in cl(A(K))$, then from (5), we get that there exists a vector z such that $A^\top z \in K^* {\setminus } sub(K^*)$ and $z^\top b \le 0$, which implies that the first part of II holds. If $b \notin cl(A(K))$, there exists a vector z such that $A^\top z \in K^*$ and $z^\top b < 0$. Since $b \in lin(cl(A(K))) = lin(A(K))$, it follows that $v^\top b = 0$ for all v such that $A^\top v \in sub(K^*)$ and, thus, $A^\top z \notin sub(K^*)$, which again implies that the first part of II holds.

Consider case 2. Since $b \notin lin(A(K))$, it follows that there exists a vector $z \in lin(A(K))^\bot = sub([A(K)]^*)$ (see (1)) such that $z^\top b \ne 0$. Thus, the second part of II holds.$\square$

Remark 9

Consider the primal-dual pair of programs (8) and (9). According to the proof of Theorem 8, if $\mathcal {P}^0 = \emptyset$, then there exists a vector $u = A^\top z \in R_{\mathcal {\tilde{D}}}$. Moreover, it can be said that

• $z^\top b \le 0$, if $b \in cl(A(K))$,

• $z^\top b < 0$, if $b \notin cl(A(K))$.

This means that, supposing that $\mathcal {D} \ne \emptyset$ and $b \notin cl(A(K))$ (which implies that the primal problem (8) is infeasible), we get that for any $(y,s) \in \mathcal {D}$, we have $\{ (y,s) + \gamma (-z,A^\top z) \ | \ \gamma \ge 0 \} \subseteq \mathcal {D}$ with $b^\top (y-\gamma z) \rightarrow +\infty$ as $\gamma \rightarrow +\infty$. Therefore, the dual problem (9) is unbounded.

Remark 10

It can be easily seen that if A is a full-rank $m\times n$ matrix $(m\le n$), i.e., the existence of the solution of $Ax=b$ is guaranteed, and the condition $\mathcal {S}(A^{\top }) \subseteq lin(K)$ holds, i.e., $\mathcal {S}(A^{\top }) \cap sub(K^{*}) = \{0\}$, then the alternatives in Theorem 8 simplify to

• (I) $\exists x\in relint(K): {A} x= b$;

• (II) $\exists z: {A}^{\top }z\in K^*{\setminus } sub(K^*)$ and $z^{\top }b\le 0$.

Moreover, for solid cones, the alternatives in Theorem 8 can be reduced to

• (I) $\exists x\in int(K): {A} x= b$;

• (II) $\exists z\ne 0: {A}^{\top }z\in K^*$ and $z^{\top }b\le 0$.

This last special case was formulated in [26] and [29] and also for the semidefinite cone in [9].

Remark 11

From (1), it follows that if $\exists x\in K: \ Ax=b$, that is, the problem (8) is feasible, then the alternatives in Theorem 8 also can be simplified as stated in Remark 10.

In this subsection, we formulate the dual counterparts of Theorems 7 and 8.

The next theorem is the “dual variant” of the generalized Farkas lemma (Theorem 7). It is formulated in [30] for linear systems and is generalized to the case of symmetric matrices and linear matrix inequalities in [25]. A similar statement is included in [11]; however, the strong alternative condition is formulated in terms of the solvability of a perturbed system.

Theorem 12

Assume that $K\subseteq \mathbb {R}^n$ is a cone satisfying Assumption 1, A is a given $m\times n$, ($m\le n$) matrix, and $b\in \mathbb {R}^m$ and $c\in \mathbb {R}^n$ are given vectors. At most, one of the following statements is true:

• (I) $\exists y: \ c-{A}^{\top }y\in K^*$;

• (II) $\exists z\in cl(K): {A}z=0$ and $c^{\top }z< 0$.

Moreover, if the cone $\mathcal {S}({A}^{\top })+K^*$ is closed, then exactly one of the statements is true.

Finally, we establish and prove a new theorem of alternatives, which deals with the relative interior of the cone $K^*$. It provides a strong alternative (and therefore also an equivalent condition) to the strict feasibility of the dual program (9).

Theorem 13

Assume that $K\subseteq \mathbb {R}^n$ is a cone satisfying Assumption 1, A is a given $m\times n$, ($m\le n$) matrix, and $b\in \mathbb {R}^m$ and $c\in \mathbb {R}^n$ are given vectors. Exactly one of the following statements is true:

• (I) $\exists y: \ c-{A}^{\top }y\in relint(K^*)$;

• (II) $\left[ \exists z\in cl(K) \setminus sub(cl(K)): {A} z= 0\ \hbox {and}\ c^{\top }z\le 0 \right]$ or $\left[ \exists z \in sub(cl(K)): Az = 0\ \hbox {and}\ c^{\top }z \ne 0 \right]$.

Proof

First, we will show that I implies $\lnot II$. Assume by contradiction that I and II hold at once. Then, $z^\top (c-A^\top \bar{y}) = \bar{z}^\top c \le 0$ for some $\bar{y}$ and $\bar{z}$ that fulfill I and II, respectively. However, from characterization (6) and (2), we obtain $\bar{z}^\top c > 0$, which is a contradiction.

Now, we will show that $\lnot I$ implies II. Suppose that $\lnot I$ holds or equivalently $\mathcal {S}(A^\top ) + relint(K^*) = relint(\mathcal {S}(A^\top ) + K^*) = \emptyset$. Regarding the vector c, there are two cases to consider.

• 1. $c \in lin(\mathcal {S}(A^\top ) + K^*) \setminus relint(\mathcal {S}(A^\top ) + K^*)$,

• 2. $c \notin lin(\mathcal {S}(A^\top ) + K^*)$.

Case 1. Since $lin(\mathcal {S}(A^\top ) + K^*) {\setminus } relint(\mathcal {S}(A^\top ) + K^*) \ne \emptyset$, it follows that $relint(\mathcal {S}(A^\top ) + K^*)$ and hence $cl(\mathcal {S}(A^\top ) + K^*)$ and $(\mathcal {S}(A^\top ) + K^*)^* = \mathcal {N}(A) \cap cl(K)$ are nontrivial. If $c \in cl(\mathcal {S}(A^\top ) + K^*)$, then from (6), we get that there exists a vector $z \in cl(K) {\setminus } sub(cl(K))$ such that $Az = 0$ and $c^\top z \le 0$, which implies that the first part of II holds. If $c \notin cl(\mathcal {S}(A^\top ) + K^*)$, there exists a vector $z \in cl(K)$ such that $Az = 0$ such that $c^\top z < 0$. Since $c \in lin(cl(\mathcal {S}(A^\top ) + K^*)) = lin(\mathcal {S}(A^\top ) + K^*)$, it follows that $c^\top z = 0$ for all $v \in \mathcal {N}(A) \cap sub(cl(K))$ and, thus, $z \notin sub(cl(K))$, which again implies that the first part of II is valid.

Case 2. Since $c \notin lin(\mathcal {S}(A^\top ) + K^*)$, from (2), it follows that there exists a vector $z \in \mathcal {N}(A) \cap sub(cl(K))$ such that $c^\top z \ne 0$, which implies that the second part of II holds.$\square$

Remark 14

Consider the primal-dual pair of programs (8) and (9). According to the proof of Theorem 13, if $\mathcal {D}^0 = \emptyset$, then there exists a vector $z \in \mathcal {N}(A) \cap cl(K)$ such that

• $c^\top z \le 0$, if $c \in cl(\mathcal {S}(A^\top ) + K^*)$,

• $c^\top z < 0$, if $c \notin cl(\mathcal {S}(A^\top ) + K^*)$.

This means that, supposing that $\mathcal {P} \ne \emptyset$ and $c \notin cl(\mathcal {S}(A^\top ) + K^*) = R_{\mathcal {P}}^*$ (which implies that the dual problem (9) is infeasible), we get that for any $x \in \mathcal {P}$ we have $\{ x+ \gamma z \ | \ \gamma \ge 0 \} \subseteq \mathcal {P}$ with $c^\top (x + \gamma z) \rightarrow -\infty$ as $\gamma \rightarrow +\infty$. Therefore, the primal problem (8) is unbounded.

Remark 15

Theorems 12 and 13 can be obtained from Theorems 7 and 8, respectively, by rewriting the alternative I using the system of linear equations $c-A^{\top }y=s$ and the cone $\mathbb {R}^m\times K^*$. For the reader’s convenience, we have included a straightforward proof of Theorem 13.

Remark 16

Analogously to the case of Theorem 8 and Remark 10, it can be seen that, requiring the condition $\mathcal {N}(A) \subseteq lin(K^{*})$ to hold (implying $\mathcal {N}(A) \cap sub(cl(K)) = \{0\}$), the alternatives in Theorem 13 can be simplified to

• (I) $\exists y: \ c-{A}^{\top }y\in relint(K^*)$;

• (II) $\exists z\in cl(K) {\setminus } sub(cl(K)): {A} z= 0$ and $c^{\top }z\le 0$.

Furthermore, if $K^*$ is solid (or cl(K) is pointed), the alternatives in Theorem 13 reduce to

• (I) $\exists y: \ c-{A}^{\top }y\in int(K^*)$;

• (II) $\exists z\in cl(K): {A} z= 0$ and $c^{\top }z\le 0$.

This last special case has been considered for the semidefinite cone in [9].

Remark 17

From (2), it follows that if $\exists y: \ c-A^{\top }y\in K^*$, that is, the problem (9) is feasible, then the alternatives in Theorem 13 also can be simplified as stated in Remark 16.

4 Strong Duality

The famous Slater result that the strict feasibility of the convex problem implies the strong duality property $d^*=p^*$ and, provided the optimal value is finite, also the existence of a dual optimal solution, is widely known. Its conic version was shown, e.g., in [11] and [31] for proper cones. In [32], the strong duality property was studied for closed and solid, but not necessarily finite-dimensional cones. Some duality results for general convex cones can be found in [20].

If one of the primal-dual pair of programs (8) and (9) is unbounded, the other is infeasible and in this trivial case $p^*=d^*$. The basic idea behind the proof of the nontrivial strong duality property is linked with the generalized Farkas lemma and its dual counterpart (Theorems 7 and 12). In the generalized version of the theorems of alternatives, the assumption of closedness of the linear image of a convex cone (or closedness of the Minkowski sum of a convex cone and a linear subspace in the dual version, respectively) is needed. However, the closedness assumption is guaranteed by the existence of the interior point in the dual (primal) feasible set. The known strong duality results for the convex conic problems are formulated in the next two theorems; see also [20] (Theorem 7) or, for conic programs with proper cones, in [11] (Theorem 2.4.1).

Theorem 18

Consider the primal-dual pair of programs (8) and (9), where the cone K satisfies Assumption 1. Then,

• (a) if $\mathcal {D}\ne \emptyset$, $d^*<+\infty$ and $\textbf{A}_c(cl(K))$ is closed, then $p^*=d^*$ and $\mathcal {P}^*\ne \emptyset$;

• (b) if $\mathcal {P}\ne \emptyset$, $p^*>-\infty$ and $\mathcal {S}(\textbf{A}_b)+(K^*\times \{0\})$ is closed, then $p^*=d^*$ and $\mathcal {D}^*\ne \emptyset$,

where $\textbf{A}_c=(A^{\top }\ c)^{\top }$ and $\textbf{A}_b=(A \ -b)$.

Recall that the proof of Theorem 18 is based on Theorem 7, Theorem 12, and the weak duality property and follows the standard scheme typically used in linear programming or the one given, e.g., in [11] for convex conic programs. The sufficient conditions that guarantee the closedness of $\textbf{A}_c(cl(K))$, and $\mathcal {S}(\textbf{A}_b)+(K^*\times \{0\})$ are $\mathcal {D}^0\ne \emptyset$ and $\mathcal {P}^0\ne \emptyset$, respectively, and the rest follows from Theorem 5. This leads us to the following statement.

Theorem 19

Consider the primal-dual pair of programs (8) and (9), where the cone K satisfies Assumption 1.

• (a) If $\mathcal {D}^0\ne \emptyset$ and $\mathcal {P}\ne \emptyset$, then $p^*=d^*$ and $\mathcal {P}^*\ne \emptyset$.

• (b) If $\mathcal {P}^0\ne \emptyset$ and $\mathcal {D}\ne \emptyset$, then $p^*=d^*$ and $\mathcal {D}^*\ne \emptyset$.

In the following example, we show that a primal-dual pair of convex programs with a nonzero duality gap can be reformulated as a primal-dual pair of convex conic programs with zero duality gap.

Example 20

Consider the convex program (see Problem 5.21 in [12])

$$\begin{aligned} \min \ {}&e^{-x_1} \nonumber \\ \text {s.t.} \ {}&\frac{x_1^2}{x_2} \le 0, \ x_2 > 0. \end{aligned}$$

(14)

It can be calculated that $p^* = 1$. The Lagrange dual function for the problem (14) takes the form

$$\begin{aligned} g(\lambda )=\inf _{x_1; x_2>0} \left\{ e^{-x_1}+\frac{\lambda x_1^2}{x_2}\right\} ={\left\{ \begin{array}{ll} 0 &{} \lambda \ge 0\\ -\infty &{} otherwise \end{array}\right. } \end{aligned}$$

Therefore, the corresponding Lagrange dual problem takes the form $\max \{ 0 \ | \ \lambda \ge 0 \}$ and, therefore, $d^* = 0$. Now, we equivalently reformulate Problem (14) as a conic problem as follows:

$$\begin{aligned} \min \ {}&t \nonumber \\ \text {s.t.} \ {}&s = 1 \\ \ {}&(x_1, x_2, t, s)^{\top } \in K_{\mathcal {F}} \nonumber \end{aligned}$$

(15)

where

$$\begin{aligned} K_{\mathcal {F}}&:= \left\{ (x_1,x_2,t,s)^{\top } \ | \ \frac{x_2}{s}> 0, \ s> 0, \ e^{-\frac{x_1}{s}} \le \frac{t}{s}, \ \frac{x_1^2}{x_2 s} \le 0 \right\} \cup \{ 0 \} \\ {}&= \left\{ (0,x_2,t,s)^{\top } \ | \ x_2> 0, \ s > 0, \ s \le t \right\} \cup \{ 0 \} \end{aligned}$$

with

$$\begin{aligned} relint(K_{\mathcal {F}}) = \left\{ (0,x_2,t,s)^{\top } \ | \ x_2> 0, \ s > 0, \ s < t \right\} . \end{aligned}$$

The corresponding conic dual problem takes the following form $\max \{z \ | \ (0, 0, 1, -z)^{\top } \in K^*_{\mathcal {F}} \}$, where

$$\begin{aligned} K^*_{\mathcal {F}} = \left\{ (u, v, w, z)^{\top } \ | \ u \in \mathbb {R}, \ v \ge 0, \ w \ge 0, \ w+z \ge 0 \right\} . \end{aligned}$$

Since $(0, 0, 1, -1)^{\top } \in \mathcal {D} \ne \emptyset$ and $(0, 1, 2, 1)^{\top } \in \mathcal {P}^0 \ne \emptyset$, according to Theorem 19(b) it holds $p^* = d^* = 1$ and, moreover, $(0, 0, 1, -1)^{\top } \in \mathcal {D}^*$. Note that the Slater condition for the convex conic programs in (15), consisting in the existence of a feasible point from the relative interior of $K_{\mathcal {F}}$, is satisfied. However, the Slater condition for convex programs (see, e.g., Section 5.2.3 in [12]) in (14) is not satisfied since the feasible set of (14) $\{ (0,x_2)^{\top } \ | \ x_2 > 0 \}$ has empty interior.

The assumptions $\mathcal {D}^0\ne \emptyset , \mathcal {P}^0\ne \emptyset$ in statements a) and b) of Theorem 19 correspond to alternative I in Theorems 13 and 8, respectively. This gives us an opportunity to combine the results and establish necessary and sufficient conditions for boundedness of the optimal solution sets $\mathcal {P}^*$ and $\mathcal {D}^*$. We obtain a new result, stated in thenext theorem.

Theorem 21

Consider the primal-dual pair of programs (8) and (9), where the cone K satisfies Assumption 1.

• (a) Assume that K is closed. The set $\mathcal {P}^{*}$ is nonempty and bounded if and only if $\mathcal {P} \ne \emptyset$, $\mathcal {D}^0 \ne \emptyset$ and $sub(R_\mathcal {P}) = \{ 0 \}$.

• (b) Suppose that $rank(A)=m$. The set $\mathcal {D}^{*}$ is nonempty and bounded if and only if $\mathcal {D} \ne \emptyset$, $\mathcal {P}^{0} \ne \emptyset$ and $sub(R_{\mathcal {\tilde{D}}}) = \{0\}$.

Proof

a) First, assume that the set $\mathcal {P}^{*}$ is nonempty and bounded. Then, clearly, $\mathcal {P}\ne \emptyset$, and we only need to show that $\mathcal {D}^0\ne \emptyset$ and ${sub(R_{\mathcal {P}})} = \{ 0 \}$. Take $x^*\in \mathcal {P}^*$ and assume by contradiction that the set $\mathcal {D}^0$ is empty. By applying Theorem 13, we obtain that:

• either there exists $z\in {{K}}{\setminus } sub({K})$ such that $Az=0$ and $c^{\top }z\le 0$ or

• there exists $z\in sub({K})$ such that $Az=0$ and $c^{\top }z <0$.

Consider the first case—then clearly, for any $\gamma \ge 0$, we have $x^*+\gamma z \in \mathcal {P}^*$. We have constructed a ray in the optimal solution set $\mathcal {P}^*$, which contradicts its boundedness. Now, consider the second case—then for any $\gamma \ge 0$, we have $x^*+\gamma z \in \mathcal {P}$, however $c^{\top }(x^*+\gamma z)<p^*$, which contradicts the optimality of $x^{*}$.

Now, assume that $\mathcal {D}^0\ne \emptyset$ and $sub(R_\mathcal {P}) \ne \{ 0 \}$. From (3), it follows that $sub(R_\mathcal {P})= \mathcal {N}(A) \cap sub({K})$. We have that there exists $0\ne z\in sub({K})$ such that $Az=0$. This time, the strong alternatives in Theorem 13 imply $c^{\top }z=0$. Again, we can construct a ray $\{x^*+\gamma z \ | \ \gamma \ge 0\} \subseteq \mathcal {P}^*$, which contradicts the boundedness of $\mathcal {P}^*$.

Conversely, suppose $\mathcal {P} \ne \emptyset$, $\mathcal {D}^0 \ne \emptyset$ and $sub(R_\mathcal {P}) = \{ 0 \}$. From Theorem 19(a) we obtain that $\mathcal {P}^*\ne \emptyset$. Assume by contradiction that $\mathcal {P}^*$ is unbounded, i.e., $\hat{x}+\gamma w \in \mathcal {P}^*\subseteq \mathcal {P}\ \forall \gamma \ge 0$. Hence, the equalities $c^{\top }w=0$ and $Aw=0$ hold, and for an arbitrary $\hat{y}\in K^*$, we have that

$$\begin{aligned} \hat{x}^{\top }\hat{y}+\gamma w^{\top }\hat{y}\ge 0, \ \forall \gamma \ge 0. \end{aligned}$$

(16)

Since the expression on the left in (16) is bounded below and $\gamma \ge 0$, it must hold $w^{\top }\hat{y}\ge 0$. Since $\hat{y}\in K^*$ was arbitrary, we get that $w\in {K^{**}= K}$. Recall that $w\in \mathcal {N}(A)$ and $c^{\top }w=0$. If $w\notin sub({K})$, then by Theorem 13, we get a contradiction with the assumption $\mathcal {D}^0\ne \emptyset$. On the other hand, $0\ne w\in sub({K})$ contradicts the assumption $\mathcal {N}(A) \cap sub({K}) = sub(R_{\mathcal {P}})= \{ 0 \}$.

b) This statement can be proved analogously, with the use of Theorem 8. The assumption $rank(A)=m$ is technical yet necessary to ensure the one-to-one correspondence between the dual variables y and s. It is only needed to argue that there would have to be a nonzero vector in $sub(R_{\mathcal {\tilde{D}}}) = \mathcal {S}(A^{\top }) \cap sub(K^{*})$ if we contradictorily assume that $\mathcal {D}^*$ is unbounded.$\square$

The assumption in Theorem 21(a) that K is closed is necessary and cannot be left out as it is shown in the following example.

Example 22

Consider the primal convex conic program in the form (8)

$$\begin{aligned} \min&\ -x_1 + x_2 \nonumber \\ \text {s.t.}&\ x_1 - x_2 = 0, \\&\ x \in K = \{(x_1,x_2)^{\top } \ | \ x_1 - x_2 > 0, \ x_1 - 2x_2 \le 0 \} \cup \{ 0 \}, \nonumber \end{aligned}$$

(17)

and the corresponding dual program in the form (9)

$$\begin{aligned} \max&\ 0 \\ \text {s.t.}&\ s = (-1-y, 1+y)^{\top } \in K^*, \\&\ K^* = \{ (s_1, s_2)^{\top } \ | \ s_1 + s_2 \ge 0, \ 2s_1 + s_2 \ge 0 \}, \end{aligned}$$

and

$$\begin{aligned} relint(K^*) = int(K^*) = \{ (s_1, s_2)^{\top } \ | \ s_1 + s_2> 0, \ 2s_1 + s_2 > 0 \}. \end{aligned}$$

We have that

$$\begin{aligned} \mathcal {P}^* = \mathcal {P} = \{ (0,0)^{\top } \}; \end{aligned}$$

thus, $\mathcal {P}^*$ is nonempty and unbounded. Obviously, $\mathcal {P} \ne \emptyset$ and $sub(R_{\mathcal {P}}) = \{ 0\}$, since K is pointed. However, $\mathcal {D}^0 = \emptyset$, and thus, Theorem 21(a) fails to hold. Note that if we replace K with cl(K) in the primal program (17), the optimal solution set will clearly be nonempty and unbounded, and thus, Theorem 21(a) will hold.

If the cone K is pointed, then $sub(R_{\mathcal {P}}) = \{ 0 \}$. Similarly, if the cone $K^*$ is pointed (i.e., the cone K is solid), then $sub(R_{\mathcal {\tilde{D}}}) = \{ 0 \}$. These special cases are covered in the following corollary. Clearly, if K is proper, then both equivalences a) and b) in Corollary 23 hold.

Corollary 23

   

• (a) Suppose K is closed and pointed. The set $\mathcal {P}^*$ is nonempty and bounded if and only if $\mathcal {P}\ne \emptyset$ and $\mathcal {D}^0\ne \emptyset$.

• (b) Suppose K is solid. The set $\mathcal {D}^*$ is nonempty and bounded if and only if $\hbox {rank}(A)=m$, $\mathcal {D}\ne \emptyset$ and $\mathcal {P}^0\ne \emptyset$.

Corollary 24

Consider the primal-dual pair of programs (8) and (9), where the cone K satisfies Assumption 1.

• (a) Suppose that K is closed. If the set $\mathcal {P}^{*}$ is nonempty and bounded, then $\textbf{A}_c(cl(K))$ is closed.

• (b) Suppose that $rank(A)=m$. If the set $\mathcal {D}^{*}$ is nonempty and bounded, then $\mathcal {S}(\textbf{A}_b)+(K^*\times \{0\})$ is closed.

Remark 25

Consider the set $\tilde{\mathcal {D}}=\{ s \ | \ (y,s)\in \mathcal {D}\}$ and the linear subspaces $sub(R_{\mathcal {\tilde{D}}})$ and $sub(R_{\mathcal {\tilde{D}}})^\bot$. Then, for $\tilde{\mathcal {D}}$, it holds that $\mathcal {\tilde{D}} = (\mathcal {\tilde{D}} \cap sub(R_{\mathcal {\tilde{D}}})^\bot ) +sub(R_{\mathcal {\tilde{D}}})$ (see [17, Proposition 1.5.4]). The authors of [20] use this fact to define the so-called normalized dual feasible set $\tilde{\mathcal {D}}_N=\mathcal {\tilde{D}} \cap sub(R_{\mathcal {\tilde{D}}})^\bot$ and the normalized dual optimal solution set as $\tilde{\mathcal {D}}^*_N=\tilde{\mathcal {D}}^*\cap sub(R_{\mathcal {\tilde{D}}})^\bot$, where $\tilde{\mathcal {D}}^*=\{ s^* \ | \ (y^*, s^*) \in \mathcal {D}^*\}$. They also study the boundedness of $\tilde{\mathcal {D}}^*_N$ and prove that $\mathcal {D}\ne \emptyset , \mathcal {P}^0\ne \emptyset$ if and only if the set $\tilde{\mathcal {D}}^*_N$ is nonempty and bounded. (See Theorem 5 in [20].) Moreover, it is easy to show that under assumption $\mathcal {P}^{0} \ne \emptyset$ it holds $sub(R_{\mathcal {\tilde{D}}})=\{0\}$ iff $\tilde{\mathcal {D}}^*=\tilde{\mathcal {D}}^*_N$. Therefore, the result of Theorem 21(b), reformulated in terms of normalized dual optimal solution set, states

• If $\mathcal {D}\ne \emptyset , \mathcal {P}^0\ne \emptyset , sub(R_{\mathcal {\tilde{D}}})=\{0\}$, then $\tilde{\mathcal {D}}^*=\tilde{\mathcal {D}}^*_N$ and it is nonemptyand bounded.

• If $\tilde{\mathcal {D}}^*$ is nonempty and bounded, then $\mathcal {D}\ne \emptyset , \mathcal {P}^0\ne \emptyset , sub(R_{\mathcal {\tilde{D}}})=\{0\}$, i.e., $\tilde{\mathcal {D}}^*=\tilde{\mathcal {D}}^*_N$.

The authors of [20] do not explicitly formulate an analogous result dealing with the normalized primal optimal solution set. The main reason is that they consider the primal conic program with a general (not necessarily closed) convex cone. However, for a closed convex cone K, we may consider the linear subspaces $sub(R_{\mathcal {P}})$ and $sub(R_{\mathcal {P}})^\bot$, and the normalized primal optimal solution set as ${\mathcal {P}}_N={\mathcal {P}}\cap sub(R_{\mathcal {P}})^\bot$. Then, the result of Theorem 21(a), reformulated in terms of normalized primal optimal solution set, states

• If $\mathcal {P}\ne \emptyset , \mathcal {D}^0\ne \emptyset , sub(R_{\mathcal {P}})^\bot = \{ 0 \}$, then ${\mathcal {P}}^*={\mathcal {P}}^*_N$ and it is nonemptyand bounded.

• If ${\mathcal {P}}^*$ is nonempty and bounded, then $\mathcal {P}\ne \emptyset , \mathcal {D}^0\ne \emptyset , sub(R_{\mathcal {P}})^\bot = \{ 0 \}$, i.e., ${\mathcal {P}}^*={\mathcal {P}}^*_N$.

More discussion is left in Appendix A.

Theorem 26

Consider the primal-dual pair of programs (8) and (9), where the cone K satisfies Assumption 1.

• (a) Assume that $\mathcal {N}(\textbf{A}_c)\cap relint(K) \ne \emptyset$. $\mathcal {D}\ne \emptyset$, $\mathcal {P}\ne \emptyset$ if and only if $p^*=d^*$, and the set $\mathcal {P}^*$ is nonempty and unbounded.

• (b) Assume that $\mathcal {S}(\textbf{A}_b^{\top })\cap relint (K^*\times \{0\})\ne \emptyset$. $\mathcal {D}\ne \emptyset$, $\mathcal {P}\ne \emptyset$ if and only if $p^*=d^*$, and the set $\mathcal {D}^*$ is nonempty and unbounded.

Proof

a) Note that the assumption $\mathcal {N}(\textbf{A}_c)\cap relint(K) \ne \emptyset$ is equivalent to the existence of a vector $v \in \mathcal {N}(A) \cap relint(K)$ such that $c^\top v = 0$.

First, assume that $\mathcal {D} \ne \emptyset$ and $\mathcal {P} \ne \emptyset$. The assumption $\mathcal {N}(\textbf{A}_c)\cap relint(K) \ne \emptyset$ is equivalent to (ii-b) in Table 1 applied to the linear map $\textbf{A}_c=(A^{\top }\ c)^{\top }$. Then, according to Theorem 5 b), the cone $\textbf{A}_c(cl(K))$ is a linear subspace (hence closed). Then, from Theorem 18(a), we get that $\mathcal {P}^*\ne \emptyset$ and $p^*=d^*$. Thus, if $x^*\in \mathcal {P}^*$ and $v \in \mathcal {N}(\textbf{A}_c)\cap relint(K)$, then clearly $x^*+\alpha v\in \mathcal {P}^*, \ \forall \alpha \ge 0$. Therefore, $\mathcal {P}^*$ must be unbounded.

Now, suppose that $\mathcal {P}^*$ is nonempty and unbounded, and it holds that $p^* = d^*$. Clearly, $\mathcal {P} \ne \emptyset$ and it remains to show that $\mathcal {D} \ne \emptyset$. Since $\mathcal {N}(\textbf{A}_c)\cap relint(K) \ne \emptyset$, we have that $\mathcal {N}(A) \cap relint(K) \ne \emptyset$ and thus (ii-b) in Table 1 holds. According to Theorem 5 b), we obtain that $\mathcal {S}(A^\top ) + K^*$ is closed. This means that the alternatives in Theorem 12 are strong: one and only one of them holds. Now, assume that $\mathcal {D} = \emptyset$, which is equivalent to $\lnot I$. It follows that II holds, and thus, there exists a vector $z \in cl(K)$ such that $Az = 0$ and $c^\top z < 0$. Take $x^* \in \mathcal {P}^*$ and $v \in \mathcal {N}(\textbf{A}_c)\cap relint(K)$. From (7), it follows that $v+z \in relint(K)$. We will show that the vector $x^*+v+z \in \mathcal {P}$. Again, from (7), we have that $x^* + v+z \in relint(K) \subseteq K$; moreover, it holds that $A(x^*+v+z) = Ax^* + A(v+z) = Ax^* = b$, thus $x^*+v+z \in \mathcal {P}$. However, we have that $c^\top (x^*+v+z) = c^\top x^* + c^\top z < c^\top x^* = p^*$, which is a contradiction with the optimality of $x^*$.

b) Note that the assumption $\mathcal {S}(\textbf{A}_b^{\top })\cap relint (K^*\times \{0\})\ne \emptyset$ is equivalent to the existence of a vector z such that $A^\top z \in relint(K^*)$ and $b^\top z = 0$.

This statement can be proved analogously, with the use of (ii-a) in Table 1, Theorem 5(a), Theorems 18(b), and 7. Note that the assumption $\mathcal {S}(\textbf{A}_b^{\top })\cap relint (K^*\times \{0\})\ne \emptyset$ implies that condition (ii-a) in Table 1 holds; moreover, it is equivalent to condition (ii-a) in Table 1 applied to the linear map $\textbf{A}_b=(A \ -b)$ and the cone $K \times \mathbb {R}$.$\square$

The following example demonstrates that the global assumption $\mathcal {N}(\textbf{A}_c)\cap relint(K) \ne \emptyset$ in Theorem 26(a) is sufficient but not necessary for the equivalence to hold: the ray defined by $v \in \mathcal {N}(A) \cap relint(K)$ in part a) may fail to exist. Similarly, the global assumption $\mathcal {S}(\textbf{A}_b^{\top })\cap relint (K^*\times \{0\})\ne \emptyset$ in Theorem 26(b) is sufficient but not necessary for the equivalence to hold: the vector z such that $A^\top z \in relint(K^*)$ in part b) may fail to exist.

Example 27

Consider the primal convex conic program in the form (8)

$$\begin{aligned} \begin{array}{rl} \min &{} x_1+x_3 \\ \mathrm {s.t.} &{} x_1+x_3= 0\\ &{} x\in {K} := \{ (x_1,x_2,x_3)^{\top } \ | \ \sqrt{x_1^2+x_2^3} \le x_3 \} \end{array} \end{aligned}$$

and the corresponding dual program in the form (9)

$$\begin{aligned} \begin{array}{rl} \max &{} 0 \\ \mathrm {s.t.} &{} s = (1-y, 0, 1-y)^{\top } \in K^{*} \\ &{} K^{*} = \{ (s_1, s_2, s_3)^{\top } \ | \ \sqrt{s_1^2+s_2^2} \le s_3 \} = K. \end{array} \end{aligned}$$

We have that

$$\begin{aligned} \mathcal {P}^* = \mathcal {P} = \{ t (-1,0,1)^{\top } \ | \ t \ge 0 \} \ne \emptyset , \end{aligned}$$

thus $\mathcal {P}^*$ is nonempty and unbounded. Moreover, it holds $p^*=d^*=0$. We alsohave that

$$\begin{aligned} \mathcal {D}^* = \mathcal {D} = \{ (1-y, 0, 1-y)^{\top } \ | \ y \le 1 \} \ne \emptyset , \end{aligned}$$

However, since $relint(K) = int(K) = \{ (x_1,x_2,x_3)^{\top } \ | \ \sqrt{x_1^2+x_2^3} < x_3 \}$, we have that $\mathcal {N}(A) \cap relint(K) = \emptyset$, which implies that there does not exist $v \in \mathcal {N}(A) \cap relint(K)$ such that $c^{\top } v = 0$.

Similarly, there is no such $z\in \mathbb {R}$ for which it holds $z(1,0,1)^{\top } \in relint(K^*)$.

If we put together results from Theorems 19 and 21, Remark 25, and Theorem 26, we can list eight sufficient conditions for strong duality property $p^*=d^*$; see Table 2.

Table 2 List of sufficient conditions for zero optimal duality gap, i.e., $p^*=d^*$

5 Conclusions

We believe that the results proposed in this paper might be useful to analyze the primal-dual relationship in specific subclasses of convex conic problems. For example, in [33], it was shown that a large class of non-convex quadratic programs with linear constraints and mixed linear and continuous variables can be represented as a completely positive programming problem. If such a program is feasible and the set of optimal solutions is nonempty and bounded (which is always the case, if the feasible set is finite, e.g., if all variables are binary), then, based on our results, it can be concluded that strong duality holds between the problem and its dual counterpart.

Our convex conic problems and the corresponding results are formulated in the way typically used in convex optimization textbooks, without more additional terminology than necessary. Our proofs are based on fundamental convex analysis and linear algebra results, which may also be useful for the readers not familiar with the topicor practitioners.

Appendix

1.1 A. More discussion

As stated in Theorems 19 and 5 in [20] (see Remark 25), the assumption $\mathcal {P}^0\ne \emptyset , \mathcal {D}\ne \emptyset$ guarantees that the sets $\tilde{\mathcal {D}}^*$ and $\tilde{\mathcal {D}}^*_N$ are nonempty. However, the boundedness of $\tilde{\mathcal {D}}^*$ is not guaranteed. This is demonstrated in the following simple example.

Example 28

Consider the primal convex conic program in the form (8)

$$\begin{aligned} \begin{array}{rl} \min &{} -5x_1 \\ \mathrm {s.t.} &{} \begin{pmatrix} 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \end{pmatrix}x= \begin{pmatrix} 2 \\ 2 \end{pmatrix}\\ &{} x\in {K} := \{ (s, t, t)^{\top } \ | \ s \in \mathbb {R}, \ t \ge 0 \} \end{array} \end{aligned}$$

and the corresponding dual program in the form (9)

$$\begin{aligned} \begin{array}{rl} \max &{} 2y_1+2y_2 \\ \mathrm {s.t.} &{} s = (-5-y_1-y_2, -y_1, -y_2)^{\top } \in K^{*} \\ &{} K^{*} = \{ (0, z_2, z_3)^{\top } \ | \ z_2 + z_3 \ge 0 \}. \end{array} \end{aligned}$$

Obviously, $\mathcal {P}^0 \ne \emptyset$, $\mathcal {D} \ne \emptyset$, A is a full-rank matrix, but ${sub(R_{\mathcal {\tilde{D}}})} = \{ (0, z_2, -z_2)^{\top } \ | \ z_2 \in \mathbb {R} \} \ne \{0 \}$. From Theorem 5, in [20], we have that $\tilde{\mathcal {D}}^{*}_N$ is nonempty and bounded. It can be calculated that $\tilde{\mathcal {D}}^{*}_N = \{ (0, 2.5, 2.5)^{\top } \}.$ However, from Theorem 21(b), we have that $\mathcal {D}^{*}$ is unbounded or empty. In fact, it is unbounded since

$$\begin{aligned} \mathcal {D}^{*} = \{ ((-5-r,r)^{\top }, (0, 5+r, -r)^{\top }) \ | \ r \in \mathbb {R} \}. \end{aligned}$$

and so is $\tilde{\mathcal {D}}^* = \{ (0, 5+r, -r)^{\top } \ | \ r \in \mathbb {R}\}$. Thus, Theorem 5 in [20] does not guarantee the boundedness of $\tilde{\mathcal {D}}^*$.