CQ-free optimality conditions and strong dual formulations for a special conic optimization problem

In this paper, we consider a special class of conic optimization problems, consisting of set-semidefinite (or Ksemidefinite) programming problems, where the set K is a polyhedral convex cone. For these problems, we introduce the concept of immobile indices and study the properties of the set of normalized immobile indices and the feasible set. This study provides the main result of the paper, which is to formulate and prove the new first-order optimality conditions in the form of a criterion. The optimality conditions are explicit and do not use any constraint qualifications. For the case of a linear cost function, we reformulate the K-semidefinite problem in a regularized form and construct its dual. We show that the pair of the primal and dual regularized problems satisfies the strong duality relation which means that the duality gap is vanishing.


Introduction
Convex optimization is a subfield of mathematical optimization where the problems of minimizing convex functions over convex sets are solved. Convex optimization has applications in a wide range of disciplines, such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, statistics, optimal experimental design, and structural optimization, where the approximation concept has proven to be efficient ( see [7,8] and the references therein). With recent advancements in computing and optimization algorithms, convex programming is indispensable in emerging methods of data mining, analytics, machine learning, as well as management, economics, and finance (e.g. [23,27,31,32]).
Conic optimization studies problems of minimizing a convex function over the intersection of an affine subspace and a convex cone. Conic models are efficiently used in dynamic processes, biomedical and chemical engineering, biology, credit risk optimization, etc. (see [3,32,35], and others). According to F. Glineur ([13]), conic optimization is an elegant framework for convex optimization, due to the fact that conic formulations present various advantages over the traditional formulations: the standard primal-dual pair of conic problems does feature a great deal of symmetry, the dual problem and the associated weak and strong duality properties can be derived in a "seamless manner" once the dual cone has been computed, and the conic formulations are helpful when designing and analyzing numerical algorithms.
The paper is organized as follows. Section 1 hosts Introduction. The problem's settings and basic notations are introduced in section 2. The new optimality conditions for a set-semidefinite problem with a polyhedral set are formulated in section 3. In section 4, we explore the properties of the set of immobile indices and the feasible set of the problem under study and use them to prove, in section 5, the main result of the paper, which is the new first-order CQ-free optimality conditions for the set-semidefinite programming (Theorem 1). In section 6, for a linear case, we deduce a regularized pair of primal and dual K-semidefinite problems and show that for this pair the strong duality property holds. The final section 7 contains some conclusions.

Problem's settings and basic notation
In what follows, we will use the following notation. Given integers p and k, let R p and R k×p denote the set of all real-valued p -vectors and k × p -matrices, respectively. Denote by R p + ⊂ R p the subset of all p -vectors with non-negative components and by S(p) the space of symmetric p × p matrices with the trace inner product: For an arbitrary nonempty set K ⊂ R p , consider the set called a cone of K -semidefinite (set-semidefinite) matrices, or simply, the K -semidefinite cone, see e.g. [11,12]. It worth mentioning that the cone C K is a generalization of a well-known and well-studied in literature cone of positive semidefinite matrices and a less studied cone of copositive matrices see e.g. [1,4,5,6] and the references therein. Indeed, for K = R p , we have C K = S + (p) and for K = R p + , we have C K = COP(p).

670
OPTIMALITY CONDITIONS AND STRONG DUALITY FOR A SPECIAL CONIC OPTIMIZATION Some properties of the cone of K-semidefinite matrices are studied in [12]. Notice, in particular, that this cone is convex for any set K.
Let A : R n → S(p) and c : R n → R be some given continuous maps. A problem is called a set-semidefinite (or K-semidefinite) optimization problem. This problem can be considered as a particular case of conic optimization problems.
In this paper, we consider problem (2) under assumptions that the map A : R n → S(p) is affine, the function c : R n → R is convex and the set K ⊂ R p is a polyhedral convex cone. Namely, we consider the following optimization problem: Problem (3) is a generalization of semidefinite and copositive programming problems since in the case K = R p , it is a problem of SDP, and in the case K = R p + , a copositive problem. The main purpose of this paper is to prove the optimality conditions and to obtain strong duality results for the set-semidefinite problem (3) that do not require any regularity conditions (constraint qualifications or, shortly, CQ). We will use here the approach developed in our previous papers (see e.g. [16,18,20]) for different classes of convex problems.

Some other formulations
It is evident that problem (3) can be considered as a special case of Semi-Infinite Programming (SIP) problems [7]. On the other hand, some more general than (3) problems can be reduced to it. In fact, consider a SIP problem in the form where the decision variable x = (x 1 , ..., x n ) is an n−vector, the index variable t = (t 1 , . . . , t p ) is a p-vector, and M ⊂ R p is a given index set. Suppose that in (4), the constraint function f (x, t) is linear w.r.t. x and linear-quadratic w.r.t. t, the cost function c(x), x ∈ R n , is convex, and the index set M is a convex polyhedron: with given matrices A ∈ R m0×p , B ∈ R m * ×p and vectors a ∈ R m0 , b ∈ R m * . Problem (4) with the index set in the form (5) is a special case of polinomial SIP problems.
Let us show that problem (4) with the index set in the form (5) and with the constraints function f (x, t) described above, can be reduced to problem (3).
Indeed, it is evident that problem (4) can be rewritten as follows: where the matrix functionĀ(x) has the formĀ(x) := n ∑ k=1Ā k x k +Ā 0 , and the problem's data is set: matricesĀ k ∈ S(p), k = 0, 1, . . . , n, D ∈ R p×n , vectors d 0 ∈ R p , q ∈ R n , and the number q 0 ∈ R.
Let X 1 be the set of feasible solutions of problem (6): Denote by d ·k ∈ R p , k = 1, . . . , n, the columns of the matrix D ∈ R p×n : D = (d ·1 . . . d ·n ), and let q = (q 1 , . . . , q n ) ⊤ . Introduce the following notation: and consider the set One can easily prove the following lemma.

Lemma 1
Given the feasible set X 1 of problem (6) and the set X 2 defined in (7), the equality X 1 = X 2 holds.
Note that the set X 2 can be considered as the feasible set of the problem in the form (3) with function A(x) replaced by B(x) and the index set K substituted by K.
It follows from Lemma 1, that the study of the special SIP problem in the form (6) with the index set (5) is equivalent to the study of the conic (set-semidefinite) problem in the form (3).

Normalized set of immobile indices
In accordance with the generally accepted definition (see, e.g., [7,13]), problem (3) satisfies the Slater condition if for somex ∈ R n it holds: t ⊤ A(x)t > 0 ∀t ∈ K \ {0}. Notice that from the very beginning, we do not make assumptions that the constraints of problem (3) satisfy the Slater condition or some other conditions.
Since the set K is a cone, problem (3) has an unbounded index set. It is easy to show that this problem is equivalent to the following SIP problem with a compact index set: where ||t|| ∞ = max k∈P |t k |. The index set T is normalized in this problem.
In our papers [15,16,19], and others, we introduced for convex SIP problems the concept of immobile indices (the indices of constraints which are active for all feasible solutions) and showed that these indices play a special role in formulating the optimality conditions which do not need the fulfillment of the Slater condition. Here, we will show how our approach can be applied to the conic problem under consideration.
Denote by X the set of feasible solutions in the SIP problem (8) and in the equivalent Ksemidefinite problem (3): and by T im the set of (normalized) immobile indices in problem (8): It can be proved (see [15]) that the Slater condition for problem (8) is equivalent to the condition T im = ∅.

OPTIMALITY CONDITIONS AND STRONG DUALITY FOR A SPECIAL CONIC OPTIMIZATION
It will be shown (see Lemma 3 below) that the set T im is either empty or a union of a finite number of convex bounded polyhedra. Then, evidently, the set conv T im is either empty or a convex polyhedron. Here conv D stays for the convex hull of a set D. In what follows, we will use the vertices of the set conv T im . Denote the set of these vertices by In the case T im = ∅, we set J = ∅.

CQ-free optimality criteria
The main aim of this paper is to prove necessary and sufficient optimality conditions for the conic optimization problem (3) and the equivalent problem (8). In this section, we formulate two theorems which have the form of criteria. These theorems do not demand that the feasible set of the problem satisfies any constraints qualifications and therefore, we refer to them as to CQ-free optimality conditions. The main concepts that allow us to formulate and prove these optimality conditions are the set of immobile indices and the vertices of its convex hull.

Theorem 1
A vector x 0 ∈ X is an optimal solution of the set-semidefinite problem (3) (and the equivalent SIP problem (8) such that where matrices U and V are defined as follows: and t * (j), j ∈ J, are vertices of the polyhedron conv T im .
We will prove this theorem in section 5.
For applications, it may be useful to reformulate Theorem 1 in a different way.
Having introduced the matrix T := (t * (j), j ∈ J) ∈ R p×m and the sets of matrices we can reformulate Theorem 1 as follows.

Theorem 2
A vector x 0 ∈ X is an optimal solution of problem (3) iff there exist matrices U 0 ∈ (C K ) * and V 0 ∈ V(K) such that At the end of this section, we would like to make several remarks.

Remark 1
It can be shown that the cone (C K ) * is dual to the cone C K .

Remark 2
Theorem 1 can be easily generalized for the case when the set K is an union of the finite number of polyhedra.
It was mentioned above that the conic problem (3) can be considered as a special case of SIP problems. Therefore, to study the optimality in this problem, one can use the optimality conditions known from the theory of SIP. Notice that almost all known optimality conditions for SIP problems require the fulfillment of some additional conditions. As a rule, these conditions are either of the Slater type (see [7,26,28,29] and the references therein) or include some weaker assumptions (see, e.g. [15,16,19]).
It worth to be mentioned that there is a small number of publications in the literature, where optimality conditions for set-semidefinite problems are formulated. These optimality conditions were also obtained under assumptions that in the case of problem (3), are equivalent to the Slater condition.
In [12], for a set-semidefinite problem in Banach and finite dimensional spaces, optimality conditions in a nondegenerate form were proved under a so-called Kurcyusz-Robinson-Zowe (KRZ) regularity condition (see (16) from [12]). For the problem in the form (3), this condition takes the form .., n}, and coneD stays for the conic hull of a set D. Let us prove that, for an affine matrix function A(x), condition (15) implies the Slater condition. Suppose that condition (15) fulfills and letS ∈ S(p) be such that t ⊤S t < 0 for all t ∈ R p , t ̸ = 0. It follows from It follows from these relations that, for problem (3), the Slater condition holds true with (x − h/α) ∈ X. Now suppose that α = 0 in (16). Then This implies that for problem (3), the Slater condition holds true with (x − h) ∈ X.
The main contribution of this paper consists of new optimality criteria formulated in Theorems 1 and 2. These criteria are CQ-free since they do not need any additional assumptions, whether it is the KRZ condition, or the Slater condition or any other CQ. This means that the scope of these optimality conditions is much wider than of other known conditions. This result was possible due to the application of the approach based on the concept of immobility. In section 6, we will show how the obtained results can be used to produce strong dual formulations for problem (3) with a linear cost function.

The properties of problem (8)
To prove Theorem 1 we need some auxiliary results which will be considered in this section.

A parametric representation of the cone K.
First, consider the cone introduced in the previous section and used to identify the index sets of the equivalent problems (3) and (8):  (18) and denote by b s ∈ R p , s ∈ S, |S| ≤ p, the vectors of an orthogonal basis of this subspace. Consider the set By construction, H * is a polyhedral convex cone. Let us show that H * is a pointed cone, i.e.
In fact, suppose that, on the contrary, there existst ̸ = 0 such thatt ∈ H * and −t ∈ H * . Then At = 0, Bt = 0, and, consequently,t ∈ H. Thus we can conclude thatt = ∑ s∈S β s b s for some scalars β s , s ∈ S. Ast ∈ H * , the following equalities should be satisfied: These equalities contradict the assumption that 0 ̸ =t = ∑ s∈S β s b s . The obtained contradiction proves that the convex polyhedral cone H * is pointed.
Denote by a s , s ∈ S * , |S * | < ∞, the extreme directions in the cone H * . Recall that by the definition, the extreme directions in a polyhedral cone are its faces of dimension one.
The following lemma formulates the known result about representation of polyhedral cones.

Lemma 2
The cone K defined in (17), can be written as follows: Note that the formula above is well consistent with the well-known fact that any nonempty convex set C admits representation in the form C = linC + (C ∩ (linC) ⊥ ), where linC is the largest subspace contained in the recession cone of C, the so-called lineality space of C (see e.g. [25], p.65). Notice that for the polyhedral cone K in the form (17), it holds linK = H and K ∩ (linK) ⊥ = H * , where H and H * are defined in (18), (19).

The properties of the set of immobile indices of problem (8)
Here, as above, we consider a general case when the set of immobile indices of problem (8) may be nonempty.

Proposition 1
Suppose that for problem (8) it is satisfied: X ̸ = ∅ and T im ̸ = ∅. Then for any x ∈ X and any t ∈ T im ⊂ K, the following relations hold true: where {b s ∈ R p , s ∈ S, |S| ≤ p} is the orthogonal basis of the subspace H defined in (18) and {a s , s ∈ S * , |S * | < ∞} is the set of extreme directions in the pointed cone H * defined in (19).
Proof. Suppose that, on the contrary, there existx ∈ X andt ∈ T im , such that one of two situations occurs: Let us assume, first, that situation 1 has happened. By construction, the vector a s0 is a recession direction fort in the cone K: t(θ) :=t + θa s0 ∈ K ∀θ ≥ 0.

Lemma 3
Given convex SIP problem (8), the immobile index set T im is either empty or can be represented as a union of a finite number of convex closed bounded polyhedra.
Proof. If T im = ∅, the lemma is evident. Suppose that T im ̸ = ∅. Consider any t ∈ T im ⊂ K. It follows from Lemma 2 that t admits the representation where cl(D) stays for the closure of a set D ⊂ R p . It is evident that By construction, for anyt ∈ T (Z), we havet ∈ T im , S + * (t) ⊂ Z, and Let us show that for anyt ∈ T (Z), we have Suppose the contrary. Then for somet ∈ T (Z), taking into account Proposition 1, there exist s 0 ∈ Z andx ∈ X, such that a ⊤ s0 A(x)t > 0. From the latter inequality and relations (23), we conclude that there exists t(ω) ∈ T im such that S + * (t(ω)) = Z and Since S + * (t(ω)) = Z and s 0 ∈ Z, the direction −a s0 is feasible for t(ω) in K, i.e. there exists θ 0 > 0 such that

OPTIMALITY CONDITIONS AND STRONG DUALITY FOR A SPECIAL CONIC OPTIMIZATION
From the latter equality and inequality (25), it follows that there exists 0 <θ ≤ θ 0 such that (t(θ)) ⊤ A(x)t(θ) < 0. But this inequality implies relations (21) that contradict the assumptionx ∈ X. Hence equalities (24) are proved. Now let us show that In fact, any vector t ∈ T (Z) can be written as follows: Taking into account this representation, equalities (24) and relations (20), we obtain Consequently, we have proved equalities (26). Given Z ∈ Z, let 0 < r = dim span( T (Z)) ≤ p, being here span(D) the linear subspace generated by D ⊂ R p . Let {t 1 , . . . , t r } ⊂ T (Z) be a basis of span( T (Z)). Denote It is easy to see that the set T (Z) is a union of a finite number of convex bounded polyhedra.
Now consider any t ∈ T (Z). By construction, we have At = 0, Bt ≥ 0, ||t|| ∞ = 1, and, consequently, t ∈ T. Since t i ∈ T (Z), i ∈ I, it follows from (26) that Taking into account these equalities, for any x ∈ X, we get

This implies that t ∈ T im and hence
Taking into account that |Z| < ∞ and the fact that each set T (Z) is a union of a finite number of convex bounded polyhedra, we conclude from the latter equality that the statement of the lemma is proved. 2 Consider the set {t * (j), j ∈ J} of vertices of the polyhedron conv T im . Recall that in the case T im = ∅, we have conv T im = ∅ and J = ∅.
Suppose that T im ̸ = ∅. Since t * (j) ∈ T im for any j ∈ J, it follows from Proposition 1 that the following relations hold true: Evidently, X ⊂ X .

Proposition 2
Consider the SIP problem (8) with the set of immobile indices T im . Let the set X be defined in (29). Then the following relations hold true: Proof.
Hence for any x, it holds Let us show that for any i ∈ J, j ∈ J, In fact, since the vector t * (i) belongs to K, it admits a representation Hence, taking into account (29), we get Inequalities (30) follow from (31) and (32). The proposition is proved. 2

The properties of the feasible set of problem (8)
In this section, we will obtain some properties of the feasible set of problem (8). These properties are closely connected with the properties of the elements of the convex hull of the set of normalized immobile indices in this problem.
where ε > 0, the set X is defined in (29), the distance between a vector l and a set B in R p is defined as ρ(l, B) = min τ ∈B ||l − τ ||, and ||a|| stays for a norm of vector a ∈ R p .

Lemma 4
Consider the SIP problem (8) with the set of immobile indices T im and the feasible set X. There exists ε 0 > 0 such that X(ε 0 ) = X, where the set X(ε) is defined in (34).
Proof. It is evident that X ⊂ X(ε) for all ε > 0. Let us show that there exists ε 0 > 0 such that X(ε 0 ) ⊂ X. Suppose the contrary. Then for all ε > 0 there exists It is easy to prove that t(ε) is also an optimal solution of the problem where T := {t ∈ K : ∥ t ∥ ∞ ≤ 1}. Notice that T ⊂ T , and the set T is convex. By construction (see Proposition 2 and definition (34)), it holds Hence we can conclude that t(ε) ∈ T (ε) \ conv T im and there exists a limiting point t * of the sequence t(ε), ε → 0 such that t * ∈ conv T im ⊂ T. To simplify notation without loss of generality suppose that t * = lim ε→0 t(ε).
Since t(ε) ∈ T ⊂ K, then t(ε) = ∑ s∈S β s b s + ∑ s∈S * α s a s , where α s ≥ 0, s ∈ S * . Consequently, taking into account the inclusion x(ε) ∈ X(ε) ⊂ X , where X is defined in (29), we have The latter inequality and inequality (35) contradict the equality (39). The lemma is proved. 2 The next lemma can be proved using the same reasoning scheme as in the proof of Lemma 3 from [20].

Lemma 5
Given SIP problem (8) with the set of immobile indices T im , for any ε > 0 there exists a vector x(ε) ∈ X such that In [24,30], a CQ-free duality theory for conic optimization was developed in terms of the so-called minimal cone. Being quite general, this theory has one disadvantage in terms of its application, namely, it is very abstract.
To the best of our knowledge, for problem (45), there are no other explicit CQ-free dual formulations satisfying strong duality relations.
We plan to dedicate a special paper to the detailed comparison of the new conic formulations obtained using the notion of the immobile indices with that from [24,30].

Conclusions
The main contribution of the paper consists in the formulation of new CQ-free optimality conditions and strong duality results for set-semidefinite programming problems. To obtain these results, we used the approach, suggested in our previous papers for convex SIP, SDP, and copositive problems, and based on the concept of immobile indices.
The main conclusions we can draw from the results of the paper, are the following.
• The concept of immobile indices being applied to set-semidefinite optimization permits one to obtain new optimality conditions. • The set of immobile indices has a special structure and its study is important. Notice that the optimality conditions formulated in Theorems 1 and 2 do not use the proper normalized immobile indices but are formulated with the help of a finite number of vertices of the convex hull of the normalized immobile index set. In the future, we plan to obtain CQ-free optimality conditions and strong duality results without explicit use of there vertices as well. • The optimality conditions and strong duality relations obtained in the paper for the special set-semidefinite problems may be used to develop numerical methods for solving these problems.