On strong duality in linear copositive programming

The paper is dedicated to the study of strong duality for a problem of linear copositive programming. Based on the recently introduced concept of the set of normalized immobile indices, an extended dual problem is deduced. The dual problem satisfies the strong duality relations and does not require any additional regularity assumptions such as constraint qualifications. The main difference with the previously obtained results consists in the fact that now the extended dual problem uses neither the immobile indices themselves nor the explicit information about the convex hull of these indices. The strong duality formulations presented in the paper for linear copositive problems have similar structure and properties as that proposed in the works by M. Ramana, L. Tuncel, and H. Wolkowicz, for semidefinite programming.

In this paper, we further develop our approach to linear copositive problems and use it to obtain a new dual problem which we refer to here as the extended dual problem. As well as the regularized dual problem from [14], the extended dual one is constructed using the notion and properties of the normalized immobile indices, but in the formulation of this problem neither these indices nor the vertices of the convex hull of the corresponding index set are present. This permitted us to formulate the extended dual problem for the linear copositive problem in an explicit form and avoid the use of additional procedures for finding the immobile indices. The new extended dual problem satisfies the strong duality relations without any CQ.
The main worthiness of the obtained results consists in the fact that the new dual formulations for Linear Copositive Programming are akin to the dual problems proposed by M.V.Ramana, L. Tuncel and H. Wolkowicz in [25,26,28] for SDP: given a linear copositive problem, we construct its dual one using the same rules as it was done in [25,26,28] for SDP problems, but with the help of different dual cones. This relation not only confirms the already known deep connection between copositive and semidefinite problems but, taking into account the impact of the duality theory developed by M. Ramana et al. for SDP, permits one to expect that the duality results proposed in this paper, will be also very promising in Linear Copositive Programming.
It is worthwhile mentioning here that at present, with exception of [14], there are no explicit strong duality formulations without CQs for Linear Copositive Progamming. All the results presented in the paper are original and cannot be obtained as a direct extension of any previous results.
The paper is organized as follows. Section 1 hosts Introduction. In Sect. 2, given a linear copositive problem, we formulate the corresponding normalized immobile index set and establish some new properties of this set. An extended dual problem is formulated in Sect. 3. We prove here that the strong duality property is satisfied. In Sect. 4, we compare the obtained duality results with that presented in [25,26,28] for SDP. It is shown that the compared dual formulations for Linear Copositive Programming and SDP are similar, being both CQ-free and providing strong duality. Although these dual formulations were obtained using different techniques, they almost coincide being applied to the class of linear SDP problems. The final Sect. 5 contains some conclusions.

Linear copositive programming problem
Here and in what follows, we use the following notations. Given an integer p > 1, R p + denotes the set of all p vectors with non-negative components, S( p) denotes the space of real symmetric p × p matrices, S + ( p) stays for the cone of symmetric positive semidefinite p × p matrices, and COP p for the cone of symmetric copositive p × p matrices The space S( p) is considered here as a vector space with the trace inner product A • B := trace (AB), for A, B ∈ S( p).
Consider a linear Copositive Programming problem in the form where the decision variable is the n−vector x = (x 1 , ..., x n ) and the constraints matrix function A(x) is defined as matrices A i ∈ S( p), i = 0, 1, . . . , n, and vector c ∈ R n are given. Problem (2) can be rewritten as follows: It is well known that the copositive problem (3) is equivalent to the following convex SIP problem: with a p -dimensional compact index set in the form of a simplex Denote by X the set of feasible solutions of problems (2)-(4), Evidently, the set X is convex. According to the definition (see e.g. [17]), the constraints of the SIP problem (4) satisfy the Slater condition if and the constraints of the copositive problem (2) satisfy the Slater condition if Here int D denotes the interior of a set D. Evidently, problem (2) (equivalently, problem (3)) satisfies the Slater condition (7) if and only if problem (4) satisfies condition (6).
Following [13,14], define the sets of immobile indices T im and R im in problems (4) and (3), respectively: It is evident that the aforementioned sets are interrelated: From the latter relations, we conclude that the set T im , the immobile index set for problem (4), can be considered as a normalized immobile index set for problem (3). In what follows, we will use mainly the set T im , taking into account its relationship with the set R im .
The following proposition is an evident corollary of Proposition 1 from [14].

Proposition 1 Given a linear copositive problem in the form (3), the Slater condition (6) is equivalent to the emptiness of the normalized immobile index set T im .
It is evident that T im = ∅ if and only if R im = {0}.
Proposition 2 Given a linear copositive problem (3), let {τ (i), i ∈ I } be some set consisting of immobile indices of this problem. Then for any x ∈ X , the following inequalities hold: Proof It is evident that for all x ∈ X , any t ∈ R im \{0} is an optimal solution of the lower level problem (see e.g. [6]) in the form Taking into account the first order necessary optimality conditions for τ (i) = (τ k (i), k ∈ P) ∈ T im ⊂ R im , i ∈ I , in the problem L L P(x), one can conclude that for all x ∈ X , it holds where e k is the k-the vector of the canonic basis of the space R p . It is evident that relations (9) imply inequalities (8).

Proposition 3 Given an index set {τ
imply the inequalities Here convS denotes the convex hull of a given set S.

Proof
The proof of the proposition is evident.
Let {τ (i), i ∈ I } ⊂ T im be a nonempty subset of the set of normalized immobile indices in problem (3). For this subset and for any ε > 0, denote where ρ(l, B) = min τ ∈B ||l − τ || is the distance between a vector l and a set B associated with the norm ||a|| = √ a a in the vector space R p . Consider the sets The following lemma is a generalization of Lemma 2 from [14]. (3), let {τ (i), i ∈ I } be a finite subset of the set of normalized immobile indices with I = ∅. Then there exists ε 0 > 0 such that X (ε 0 ) = X, the set X (ε) being defined in (14) with the set T (ε) as in (12).
It should be noticed that the lemma above can be considered as a generalization of Lemma 2 from [14] since it is formulated for an arbitrary subset {τ (i), i ∈ I } of T im , while in Lemma 2 from [14] we considered the fixed subset of T im , namely, the set of vertices of convT im .

An extended dual problem for Linear Copositive Programming
In this section, we will discuss some dual formulations of the linear copositive problem (2).
Given an arbitrary cone K ∈ S( p), the corresponding dual cone K * is defined as It is known that the cone of symmetric positive semidefinite matrices S + ( p) is self-dual, i.e. S * + ( p) = S + ( p) but the cone of symmetric copositive matrices COP p defined in (1), is not.
It can be shown (see e.g. [3]) that given the cone COP p , its dual cone CP p is the cone of so-called completely positive matrices: Then the (standard) Lagrangian dual problem for (2) can be written as follows [2]: It is well known (see [2] Theorem 3.1) that if the constraints of problem (2) satisfy the Slater condition, then there is no gap between the optimal values of problems (2) and (18).
If the constraints of problem (2) do not satisfy the Slater condition, then the positive gap is possible (see an example at the end of this section).
The aim of this section is to formulate an extended dual problem for problem (2) and show that there is no duality gap for this pair of dual problems.
For a given finite integer m 0 ≥ 0, consider the following problem: where Notice that in the case m 0 = 0, we consider that the index set {1, ..., m 0 } is empty and the constraints are absent in problem (19). Hence, for m 0 = 0, problem (19) takes the form (18): (2) and

Lemma 2 [Weak duality] Let x ∈ X be a feasible solution of the primal linear copositive problem
be a feasible solution of problem (19). Then the following inequality holds: exists a matrix B m with non-negative elements in the form The matrix B m above is composed by the blocks containing some matrices where Here and in what follows, |I | denotes the number of elements in a set I .
Consider the first group of constraints of the dual problem (19): Due to (22) and (23), these constraints can be rewritten in the form It follows from (24) that for any x ∈ R n , we have Since the inequalities should be fulfilled, equality (25) Thus, one can conclude that τ 1 (i) ∈ T im , i ∈ I 1 , and, consequently (see Proposition 2), it holds Suppose that for some m ≥ 1, it was shown that Due to (22) and (23), the constraints (U m+1 + W m ) • A j = 0, j = 0, 1, ..., n, of problem (19) can be rewritten as follows: It follows from the latter equalities that for any x ∈ R n , we have i∈I m+1 By the hypothesis above, inequalities (28) are satisfied. Then, taking into account that λ m (i) ∈ R p + , i ∈ I m , and for any x ∈ X , inequalities (26) hold, we conclude from (29) that for any x ∈ X , the following equalities are valid: Hence, τ m+1 (i) ∈ T im , i ∈ I m+1 , and, according to Proposition 2, it holds Replace m by m + 1 and repeat the considerations for all m < m 0 .
Let m = m 0 . In this case, relations (28) have the form of problem (19) can be represented as follows: Then, evidently, From (26) and (30), we conclude that These inequalities together with equality (32) imply (21). The lemma is proved. (2) have an optimal solution. Then there exist a number 0 ≤ m 0 < ∞ and a feasible solution

Lemma 3 [Strong duality] Let problem
of problem (19) such that for any optimal solution x 0 of problem (2), it holds Proof To prove the lemma, we will algorithmically construct the number m 0 and the matrices (33). Iteration # 0. Consider the following SIP problem: with the set T defined in (5). If there exists a feasible solution (x,μ) of this problem with μ < 0, then set m 0 := 0 and GO TO the Final step.
Otherwise for any x ∈ X , the vector (x, μ 0 = 0) is an optimal solution of problem (35). It should be noticed that in problem (35), the index set T is a compact set, and the constraints of this problem satisfy the Slater condition. Hence, based of Propositions 5.105 and 5.107 from [6], we conclude that there exist indices and numbers It follows from (36) that the set I 1 is nonempty and Let us sum up all these equalities. As a result we get Since γ (i) > 0, i ∈ I 1 and inequalities (26) should hold true, then it follows from (37) that By definition, this means τ (i) ∈ T im , i ∈ I 1 .
Let us set Denote T 1 := conv {τ (i), i ∈ I 1 } and proceed to the next iteration.
where ε 1 > 0 is such a number that the set of feasible solutions of problem (39) with μ = 0 coincides with the set X of feasible solutions of problem (2). According to Lemma 1, such ε 1 > 0 exists. If there exists a feasible solution (x,μ) of problem (39) withμ < 0, then STOP and GO TO the Final step with m 0 := 1.
Otherwise, (x, μ 0 = 0) with any x ∈ X is an optimal solution of problem (39). In the SIP problem (39), the index set {t ∈ T : ρ(t, T 1 ) ≥ ε 1 } is compact, and the constraints satisfy the following Slater type condition: Hence (see Theorem 1 and its Corollary from [18])), there exist indices and numbers It follows from (42) that ΔI 1 = ∅. Let us show that for all x ∈ X , it holds In fact, from (41) it follows Let us sum up these equalities. As a result, we get We have proved above that τ (i) ∈ T im , i ∈ I 1 . Then, from Proposition 2 it follows Equalities (43) are the consequence of inequalities (26), (45) and equalities (44). From (43), it follows τ (i) ∈ T im , i ∈ ΔI 1 , and from (36) and (41), we get where I 2 := I 1 ∪ ΔI 1 .
To prove the finiteness of the algorithm described in this lemma, we will use a specific procedure that modifies the index set {τ (i), i ∈ ΔI } using the index set {τ (i), i ∈ I } , where I and ΔI are the sets which we have at the current iteration #m of the algorithm: I := I m , ΔI := ΔI m .
The main aim of the procedure is as follows: having fixed the index set {τ (i), i ∈ I }, (found at the previous iteration #(m − 1)), we replace the indices τ (i) ∈ T im , i ∈ ΔI , (found at the current iteration #m ) by the new onesτ (i) ∈ T im , i ∈ ΔI , so as to satisfy the inclusion for all i ∈ ΔI , and meet the relations Moreover, we want that the indices from the fixed index set {τ (i), i ∈ I }, together with new onesτ (i) ∈ T im , i ∈ ΔI , satisfy relations similar to (46). For this purpose we will modify some coefficients γ (i), i ∈ I ΔI , and λ(i), i ∈ I .
On the first iteration of the algorithm, we apply the procedure to the data set Procedure DAM (Data Modification).
The Procedure starts with an initial data set such that then STOP. The Procedure DAM is complete. If (50) is not satisfied, then find s 0 ∈ ΔI and i 0 ∈ I such that Let us find indexτ (s 0 ) that has the formτ (s 0 ) = τ (s 0 ) − θτ (i 0 ) and satisfies the relationŝ For this purpose, set θ := min k∈P θ k > 0, where It is easy to verify that the constructed indexτ (s 0 ) satisfies (52). Let us show that θ < 1.
It is easy to verify that relations (49) hold true with the modified data set. For the modified data set, check condition (50). If it is satisfied, then STOP, the procedure is complete. If (50) is not satisfied, then find new indices s 0 ∈ ΔI and i 0 ∈ I such that inclusion (51) is valid and repeat the steps described above.
The Procedure DAM is completely described.
Let us go back to the proof of the lemma. Recall that we are performing the Iteration #1 of the algorithm. Having applied the Procedure DAM to the data set (47), one obtains a new (modified) data set in the same form (47) such that • the indices τ (i), i ∈ I 1 , are the same as in the initial data set (i.e., the procedure left these indices unchanged); • the modified indices τ (i), i ∈ ΔI 1 , are the immobile ones in problem (3); • for the modified indices τ (i) and numbers γ (i), i ∈ ΔI 1 , relations (40) are fulfilled; • for the modified vectors λ 1 (i) and numbers γ (i), i ∈ I 1 , it holds λ 1 (i) ∈ R p + , γ (i) > 0, i ∈ I 1 ; • for the modified data set (47), relations (50) with ΔI = ΔI 1 , I = I 1 and (46) are satisfied.
Using the new data (obtained as the result of applying the Procedure DAM to the initial data set (47)), denote: . Then relations (46) can be written as follows: GO TO the next iteration.
Iteration # m, m ≥ 2. By the beginning of this iteration, the numbers β m (i) > 0, i ∈ I m , as well as the indices, vectors and numbers are found such that Using these data, matrix was constructed. Denote T m := conv{τ (i), i ∈ I m } and consider the problem where ε m > 0 is such a number that the feasible set of problem (56) with μ = 0 coincides with the feasible set X of problem (2). According to Lemma 1, such ε m exists. If there exists a feasible solution (x,μ) of problem (56) withμ < 0, then STOP and GO TO the Final step with m 0 := m.
Otherwise for any x ∈ X , vector (x, μ 0 = 0) is an optimal solution of problem (56). Since in problem (56) the index set {t ∈ T : ρ(t, T m ) ≥ ε m } is compact, and the constraints satisfy the Slater type condition, then the optimality of (x, μ 0 = 0) provides that there exist indices and numbers and vectors that satisfy the following equalities: By analogy with how this was done at Iteration # 1, we can show that equalities (59) imply the equalities and, therefore, τ (i) ∈ T im , i ∈ ΔI m . Based on (54) and (59), one can conclude that i∈I m+1 where I m+1 := I m ∪ ΔI m , and the vectors λ m (i) ∈ R p + , i ∈ I m , are constructed as follows: Having applied the Procedure DAM to the data set one will get the modified data set (in the same form) such that Using these new data, let us set where matrix V 0 m was defined at the previous iteration according to (55). Then relations (60) can be written in the form: Perform the next Iteration #(m + 1). Final step. It will be proved in Lemma 4 (see below) that the algorithm consists of a finite number of iterations.
In situation a) the constraints of the original problem (2) satisfy the Slater condition. Hence, according to the well-known optimality conditions (e.g. see the KKT condition (4) in [2]), if x 0 is an optimal solution of problem (2), then there exists a matrix U 0 ∈ CP p such that It follows from the relations above that U 0 is a feasible solution of the dual problem (20) and equality (34) holds.
Consider situation b): m 0 > 0. By the beginning of the final step, the matrices where ε m 0 > 0 is the number used when problem (56) with m = m 0 was formulated. The way the number ε m 0 has been chosen guarantees that the feasible set of problem (63) coincides with the feasible set of problem (2). Problem (63) satisfies a Slater type condition since, by construction, and {t ∈ T : ρ(t, T m 0 ) ≥ ε m 0 } is a compact set. Let x 0 be an optimal solution of problem (2). Then vector x 0 is optimal in problem (63) as well. Hence, there exist indices, numbers and vectors Let us set . Then relations (64) take the form It follows from (38), (53), (62), and (66) that the constructed set of matrices (33) is a feasible solution of problem (19).
It was shown above that for any feasible solution x ∈ X of problem (2) and any feasible solution (20) of problem (19), the inequality (21) holds. From the equalities (32) and (65), it follows that the feasible solution x 0 ∈ X of the primal problem (2) and the feasible solution (33) of problem (19) constructed above, turn the inequality (21) into equality. The lemma is proved.

Lemma 4 The algorithm, described in the proof of Lemma 3, is finite (i.e. it stops after a finite number of iterations).
Proof If the algorithm has stopped on the Iteration # 0 or the Iteration # 1, the lemma is proved.
Otherwise, let us consider an Iteration # m of the algorithm for some m ≥ 2 . At the beginning of this iteration, we have the set of indices τ (i) ∈ R p + , i ∈ I m , where As before, denote P + (i) := {k ∈ P : τ k (i) > 0}, i ∈ ΔI s , s = 0, 1, ..., m − 1.
For any k, 2 ≤ k ≤ m, and any s, 0 ≤ s ≤ k − 2, by construction, it holds and the relations (50) are fulfilled with ΔI = ΔI k−1 and I = I k−1 . Hence wherefrom we conclude Consequently, all the sets P + (i s ), s = 0, 1, ..., m − 1, are different. Taking into account that on each Iteration # m it holds ΔI s = ∅, s = 0, 1, ..., m −1, one can conclude that the number m 0 of the iterations fulfilled by the algorithm, cannot be greater than some finite number m * , where m * is the maximal number of all different subsets of the set P satisfying (67). The lemma is proved.

Remark 1
The main contribution of the algorithm used in the proof of Lemma 3, consists in the justification of the existence of a finite number m 0 and the corresponding feasible solution (33) of problem (19) for which equality (34) is satisfied. It is worthwhile mentioning that it was not the aim of this paper to find a "good" estimate of the minimal value of the number m 0 .
Notice also that it is possible that someone can offer other (pehaps more complex) procedures for finding the finite sets of matrices (33) satisfying the constraints of problem (19) and the equality (34). Some of such procedures may provide a better (smaller than m * ) estimate of the number m 0 . It easy to check that It follows from the latter relations that the constructed set of matrices (U 1 , W 1 , D 1 ; U ) is a feasible solution of problem (19) with m 0 = 1 and the corresponding value of the dual cost function is equal to zero. Taking into account the weak duality inequality (21) and the equality c x 0 = 0, we conclude that the feasible solution (70) is optimal for the extended dual problem (19) with m 0 = 1. Consequently we have shown that the extended dual problem has an optimal solution and there is no positive duality gap between the original primal problem and its extended dual.
At the end of this section, we would like to note that as far as we know, with the exception of the paper [14] mentioned above, all previously published optimal conditions and duality results for Linear Copositive Programming are formulated under the Slater condition. In our research, we do not suppose that the Slater condition is satisfied. All this demonstrates the importance and novelty of the results of the paper.

A comparison of the obtained duality results for the linear copositive problem with dual formulations for the linear SDP
The main goal of this section is to show that for the given linear copositive problem (2), its extended dual problem (19) is constructed here using the same rules which were used in [25,26,28] to built a dual problem for SDP but with the help of different dual cones. Consider a linear SDP problem Following [26], let us adduce the M. Ramana et al.' extended dual for this problem: It is easy to notice that the new dual problem (19) obtained in this paper for problem (2), has a similar structure and properties as the dual problem (ED-R) for SDP problem (71). Nevertheless, it is worth mentioning that these dual problems were obtained using different approaches: the dual problem (19) was formulated and its properties were established using (implicitly) the concept of the immobile indices while the dual SDP problem (ED-R) (referred in [26] as the regularized dual problem (DRP)) was derived using the notion of the minimal cone which was described there as the output of a special procedure.
To show that there exists more closed relationship between copositive and SDP dual formulations, we will formulate for the SDP problem (71), a new dual problem which is a little different from the formulation (ED-R).
For this purpose, we apply to the SDP problem (71), the approach developed hereby for Linear Copositive Programming. Having repeated the process of building the dual problem as it was described in Sect. 3, one can obtain the extended dual to problem (71) in the form (U m + W m−1 ) • A j = 0, j = 0, 1, ..., n, m = 1, ..., m 0 , The only difference in formulations (ED) and (ED-R) consists of the right lower blocks of the matrices (72) Here μ max (Q) denotes the maximal eigenvalue of a matrix Q ∈ R p× p . It is easy to check that, by construction, we have Let us show that or equivalently, It is known (see [21], p. 230) that for any real symmetric matrix Q ∈ S( p), the inequality t Qt ≤ μ max (Q)t t is satisfied for any t ∈ R p . Hence Inclusion (76) is proved.
Suppose that for some m ≤ m 0 we have constructed matrices and a number ρ(m) > 0 such that W m−1 = ρ(m)W m−1 and the following relations hold: Let us set Applying the rules described above for the cases where m = m 0 , m 0 − 1, ..., 2, we can construct matrices U , U m 0 , ..., U 2 , W m 0 , ..., W 2 , W 1 , and the number ρ(2) > 0 such that Set ρ(1) := max{1, ρ 2 (2)μ max (L 1 L 1 )}, U 1 := ρ(1)U 1 . One can check that the matrices constructed above, form a feasible solution (75) of problem (ED-R) and it holds Hence, for the SDP problem (71), we have shown that the dual problem in the form (ED) is a slight modification of the known dual problem (ED-R). Now, let us compare two pairs of primal and dual problems: (α) the linear copositive problem (2) and its dual one (19), and (β) the SDP problem (71) and its dual one (ED).
One can see that these pairs of dual problems are constructed in the spaces S( p) and S(2 p) using the same rules, but their constraints are defined with the help of different dual cones: -in the pair of problems (2) and (19), the cone COP p is used to formulate the constraints of the primal copositive problem and the dual cones CP p and CP 2 p are used to formulate the constraints of the dual one; -in the pair of SDP problems (71) and (ED), the cone S + ( p) is used to formulate the constraints of the primal SDP problem and the dual cones S + * ( p) = S + ( p) and S + * (2 p) = S + (2 p) are used for the dual formulation.
This similarity points to a deep relationship between these two classes of conic problems, Linear Copositive Programming and SDP. At the same time, it is worth mentioning that copositive problems are more complex and less studied when compared with that of SDP.

Remark 4
When comparing the complexity of the procedures described above for constructing the pairs of dual problems in SDP and Linear Copositive Programming, notice the following.
-For SDP problems, one has an estimate m 0 ≤ min{n, p} of the number m 0 . This estimate can be found using the fact that the set of immobile indices for an SDP problem is a subspace of R p and the properties of positive semidefinite matrices are well-studied. -For linear copositive problems, determining a good estimate of m 0 is a much more challenging task as the set of immobile indices is a union of a finite number of convex cones in R p . Notice that the cone of copositive matrices and its dual cone (the cone of completely positive matrices) are not so well studied (there are many open questions here [5,9]). -The cones of copositive and completely positive matrices are neither self-dual nor homogeneous (see [10]).
As it was noticed above, finding a good estimate of the number m 0 for copositive problems was not our purpose here. We plan to devote a special paper to this issue.

Conclusions and future work
The main contribution of the paper consists in developing a new approach to dual formulations in Linear Copositive Programming. This approach permitted us to formulate a new extended dual problem in explicit form and to close the duality gap between the optimal values of the copositive problem and its extended dual without any CQs or other additional assumptions.
To the best of our knowledge, with the exception of our previous papers [14,17], in Linear Copositive Programming, there are no other known explicit strong dual formulations that do not require CQs.
In [14,17], the dual problems were formulated based on the explicit knowledge of the immobile index set. The advantage of the dual results presented in this paper if compare with the results mentioned above, consists in the fact that now there is no need to find explicitly either the elements of the normalized immobile index set or the extremal points of its convex hull. For linear copositive problems, the dual formulation obtained in the paper is original and different from that published before.
The new dual formulation for Linear Copositive Programming is similar to the dual formulation for SDP problem proposed by M.Ramana et al. in [26]. This similarity and the fact that the duality results obtained in this paper (i) do not use CQs, (ii) have explicit formulation, and (iii) are strong, motivate us to study other applications of the developed approach based on the notion of the immobile indices.
In our future work, we are going to find a better estimate of the number m 0 that is essential for our dual formulation. To obtain this estimate, it will be necessary to study new properties of the extended dual problem and its feasible set. We plan also to apply the results of the paper for other classes of copositive problems with the aim to develop new explicit optimality conditions.