Algorithmic determination of immobile indices in convex SIP problems with polyhedral index sets

Abstract The concepts of immobile indices and their immobility orders are objective and important characteristics of feasible sets of semi-infinite programming (SIP) problems. They can be used for the formulation of new efficient optimality conditions without constraint qualifications. Given a class of convex SIP problems with polyhedral index sets, we describe and justify a finite constructive algorithm (algorithm DIIPS) that allows to find in a finite number of steps all immobile indices and the corresponding immobility orders along the feasible directions. This algorithm is based on a representation of the cones of feasible directions in the polyhedral index sets in the form of linear combinations of extremal rays and on the approach proposed in our previous papers for the cases of immobile indices’ sets of simpler structures. A constructive procedure of determination of the extremal rays is described, and an example illustrating the application of the DIIPS algorithm is provided.


Introduction
Semi-infinite programming (SIP) deals with extremal problems that involve infinitely many constraints in a finite dimensional space. Due to numerous theoretical and practical applications, today semi-infinite optimization is a topic of a special interest (see Hettich and Kortanek 1993;Bonnans and Shapiro 2000;Stein 2003;L opez and Still 2007; and the references therein). The most efficient methods for solving optimization problems are usually based on optimality conditions that permit not only to test the optimality of a given feasible solution, but also to find better directions to optimality. Usually, the optimality conditions are formulated for certain classes of optimization problems. This allows for effective use of specific structures of the problems under consideration (see e.g. Bonnans and Shapiro 2000;Stein 2003).
In our papers (see e.g. Kostyukova et al. 2008;Kostyukova and Tchemisova 2014), we introduce the concepts of immobile indices and their immobility orders for different classes of SIP problems. For a given SIP problem, immobile indices can be defined as indices of the constraints that are active for all feasible solutions. It was shown that immobile indices are objective and important characteristics of feasible sets allowing to formulate efficient optimality conditions which do not use constraint qualifications (CQ) and can be successfully applied for building new constructive numerical methods. We proved optimality conditions for different classes of convex SIP problems, having formulated these conditions in both implicit and explicit forms. Obviously, to use these optimality conditions, as well as to develop numerical methods based on these conditions, it is necessary to have practical algorithms that determine the immobile indices and their immobility orders. In our paper Kostyukova et al. (2008), we described such an algorithm for the case when the indices of the constraints have dimension one (the index set is a compact on the line). This algorithm cannot simply be generalized for the case of multidimensional index sets, because in this case the feasible sets have a more complex structure and, in turn, should be represented constructively in terms of their extremal rays.
In this paper, we describe and justify a finite constructive algorithm (algorithm DIIPS) that determines immobile indices in convex SIP problems with polyhedral index sets. Given a feasible solution and the corresponding set of active indices, we describe the conforming cones of feasible directions in terms of the extremal rays. These rays are determined using a procedure that was specially elaborated for this purpose. Next, we use the DIIPS algorithm to find the set of immobile indices of the SIP problem and the corresponding immobility orders along the extremal directions. An example illustrating application of the DIIPS algorithm is provided.

Immobile indices and immobility orders in SIP problems with polyhedral index sets
Consider a convex SIP problem in the form where K is a finite index set; the objective function c(x) and the constraint functions f ðx; tÞ; t 2 T; are convex w.r.t. x 2 R n ; vectors h k 2 R s and numbers Dh k ; k 2 K are given. Notice that here the index set T is a convex polyhedron. Denote by X the feasible set of problem (P): X: ¼ fx 2 R n : f ðx; tÞ 0 8t 2 Tg: Given t 2 T, denote by K a ðtÞ & K the corresponding set of active indices: and by L(t) the corresponding set of feasible directions in T: Let T a ðxÞ & T be the set of active in x 2 X indices: T a ðxÞ :¼ ft 2 T : f ðx; tÞ ¼ 0g: Denote by T Ã the set of all immobile indices in (P). It is evident that T Ã & T a ðxÞ for all x 2 X.
Definition 2.2. The constraints of problem (P) satisfy the Slater condition if there exists x 2 R n such that f ð x; tÞ<0; t 2 T: In Kostyukova and Tchemisova (2012), it is proved that the convex SIP problem (P) with X 6 ¼ ; satisfies the Slater condition (the Slater CQ) if and only if the set of immobile indices in this problem is empty. Therefore, the emptiness of the set T Ã can be considered as a CQ which is equivalent to the Slater CQ.
The following definition determines important quantitative characteristics of the immobile indices.
Definition 2.3. Given an immobile index t 2 T Ã and a nontrivial feasible direction l 2 Lð tÞ, let us say that t has the immobility order qð t; lÞ along l if 1. d i f ðx; tþa lÞ da i j a¼þ0 ¼ 0 8x 2 X; i ¼ 0; :::; qð t; lÞ; 2. there exists a feasible x ¼ xð t; lÞ 2 X such that d ðqð t ; l Þþ1Þ f ð x; t þa lÞ da ðqð t ; l Þþ1Þ j a¼þ0 6 ¼ 0: Here and in what follows, we consider the set fi ¼ s; :::; kg to be empty if k<s: We denote d 0 f ðx; t þa lÞ da 0 j a¼þ0 :¼ f ðx; tÞ: Notice that Definition 2.3 can be easily generalized for all indices t 2 T if one sets qðt; lÞ :¼ À1 for any t 2 T n T Ã ; l 2 LðtÞ: 3. The cone of feasible directions in the case of a polyhedral index set Given the convex SIP problem (P), consider an index t 2 T: Denote by L :¼ Lð tÞ the set of feasible in t directions that is defined in (3). Evidently, L is a polyhedral cone in R s and hence it is finitely generated by some vector set in R s . In this section, we will present constructive rules for finding this vector set.

Representation of the set L in terms of the extremal rays
Given t 2 T; consider the corresponding set K :¼ K a ð tÞ and the set D L & R s defined as follows: K Þ: Set p :¼ sÀm and suppose that the set b i ; i ¼ 1; :::; p f g is a basis of D L. Consider the set D L ¼ L \ D L ? ; where D L ? is the orthogonal complement of the subspace D L in R s . One can easily prove that the set D L is a pointed cone, i.e. it is a cone with the following property: for any l 6 ¼ 0 it holds: l 2 D L ) Àl 6 2 D L: Then, there exists a finite set of vectors such that the cone L can be represented in the form where vectors b i ; i 2 f1; :::; pg, are defined in (5) and b i 2 R; i 2 f1; :::; pg. Therefore, for any t 2 T there exist (finite) sets of vectors (5) and (6) such that the set of feasible directions in t can be represented in the form (7). The vectors of the sets (5), (6) are usually referred to as extremal rays, vectors (5) being bidirectional and vectors (6) being unidirectional rays.

The rules for constructing the extremal rays
In the literature, different methods for constructive representation of polyhedral cones are proposed (see e.g. Chernikova 1968;Fernandez and Quinton 1988). Here, we describe a simple procedure that can be used for the determination of the sets of extremal rays (5) and (6) and, therefore, for the representation of the set L in the form (7). Given k 2 K ; consider vector h k defining the index set of the problem (P). Denote by h ki ; i 2 S :¼ f1; 2; :::; sg, the components of this vector: h T k ¼ ðh ki ; i 2 SÞ: Let H be a j K j Â jSjÀ matrix composed by these components: is not singular: detðH 0 Þ 6 ¼ 0. By construction, H 0 is a square sub-matrix of the matrix H such that rankH ¼ rankH 0 ¼ m.

Construct vectors
whose components are defined by the following rules: It is easy to verify that vectors (9) form a basis of the space KerH ¼ D L. Therefore, we can set in (5) that fb i ; i ¼ 1; :::; pg ¼ f b i ; i 2 SnS 0 g: Consider vector h 0 :¼ P k2 K h k . If h 0 ¼ 0 2 R s , then the set of vectors (6) in (7) is empty.
Suppose that h 0 6 ¼ 0. Denote by X the set of subsets N Ã & K such that jN Ã j ¼ mÀ1 and detðDðN Ã ÞÞ 6 ¼ 0; where DðN Ã Þ ¼ ðh 0 ; h k ; k 2 N Ã ; b i ; i ¼ 1; :::; pÞ T 2 R sÂs : Given N Ã 2 X; let aðN Ã Þ be the first column of the matrix ÀD À1 ðN Ã Þ, i.e. aðN Ã Þ ¼ ÀD À1 ðN Ã Þe 1 . Set X Ã :¼ fN Ã 2 X : h T k aðN Ã Þ 0; k 2 K nN Ã g: It can be easily verified that for representation (7) we can choose the set of vectors a i ; i 2 I, in the form Remark 1. From the constructions above, it follows that in the case m ¼ jS 0 j ¼ j K j, we have I ¼ f1; :::; mg, and the vectors a i ¼ ða ij ; j 2 SÞ; i 2 I, can be constructed by the following rule: a ij ¼ 0; j 2 SnS 0 ; ða ij ; j 2 S 0 Þ T ¼ ÀH À1 0 e i ; i ¼ 1; :::; m; where e i 2 R m is the i-th vector of the canonic basis of R m , and the matrix H 0 is given in (8).
Remark 3. As it was noted above, the set fa i ; i 2 Ig is empty (I ¼ ;) when h 0 ¼ 0. It can be proved that h 0 6 ¼ 0 if the interior of the polyhedral index set T is not empty, i.e. if the constraints defining T satisfy the Slater condition 9t 2 T : h T kt < Dh k 8k 2 K:

Immobile indices and CQ-free optimality conditions
Assumption 4.1. Suppose that given a convex SIP problem in the form (P), it holds: X 6 ¼ ;, the set T is bounded, and q t; l ð Þ 1; 8t 2 T Ã ; 8l 2 L t ð Þ n 0 f g: We consider here that conditions (11) are trivially fulfilled if T Ã ¼ ;: The following result is proved in Kostyukova and Tchemisova (2014) (Proposition 3.1).
Proposition 4.2. Assumption 4.1 implies that the set of immobile indices T Ã consists of a finite number of elements: T Ã ¼ ft Ã j ; j 2 J Ã g with some finite index set J Ã , and there exists x 2 X such that jT a ð xÞj<1: For any immobile index t Ã j ; j 2 J Ã , consider the corresponding cone of feasible directions Lðt Ã j Þ defined in (3). Denote by b i j ð Þ; i ¼ 1; :::; p j ; a i j ð Þ; i 2 I j ð Þ; j 2 J Ã ; the extremal rays generating this cone. These rays can be found by the rules described in the previous section. Denote It can be proved that under Assumption 4.1, the optimality conditions for problem (P) have the form of the following criterion (Theorem 3.2 in Kostyukova and Tchemisova (2014)).
Theorem 4.3. Let Assumption 4.1 be fulfilled for the convex SIP problem (P). Then a vector x 0 2 X is optimal in this problem if and only if there exists a finite set of indices ft j ; j 2 J a ðx 0 Þg & T a ðx 0 Þ n T Ã where jJ a ðx 0 Þj n; such that the vector x 0 is optimal in the following auxiliary problem: Theorem 4.3 gives optimality conditions for problem (P) in the form of an implicit optimality criterion and uses the information about the immobile indices and the extremal rays representing the corresponding cones of feasible directions. In Kostyukova and Tchemisova (2014), these conditions were reformulated in different forms, including explicit optimality conditions. All these conditions are CQ-free and more efficient when compared with other optimality conditions known from the literature.
It is evident that to apply Theorem 4.3 one should know the set of immobile indices T Ã and the corresponding index sets (13). In the next section, we present a constructive algorithm for determination of the set of immobile indices and the corresponding sets (13) for problem (P). We call this algorithm DIIPS since it Determines the Set of Immobile Indices in SIP problems with Polyhedral index Sets.

Algorithm DIIPS
Given a convex SIP problem in the form (P), suppose that Assumption 4.1 is satisfied. It follows from Proposition 4.1 that there exists a feasible solution x 2 X such that T a ð xÞ ¼ f t j ; j 2 J g with j J j<1: Suppose here that such a feasible solution x 2 X, the corresponding set T a ð xÞ ¼ f t j ; j 2 J g; the vectors b i ðjÞ; i ¼ 1; :::; p j ; a i ðjÞ; i 2 IðjÞ; defining the cones Lð t j Þ; j 2 J ; and the index setsĨðjÞ ¼ fi 2 IðjÞ : @f T ð x; t j Þ @t a i ðjÞ ¼ 0g; j 2 J ; are known.
Initializing. Set J ð0Þ Ã :¼ ;: General iteration. We start the ðk þ 1Þ-st iteration of the algorithm (k ! 0) having the following sets constructed on the previous iteration: Notice that at the first iteration (k¼ 0) we do not use the sets I ðkÞ 0 ðjÞ &ĨðjÞ; j 2 J ðkÞ Ã ; since the set J ð0Þ Ã is empty.
; (14) where for l 2 R s ; jjljj ¼ max i¼1;:::;s jl i j: Here BðjÞ :¼ ðb i ðjÞ; i ¼ 1; :::; Let us consider the following set: It can be shown that x 2 X ðkþ1Þ . For all j 2 J nJ ðkÞ Ã , consider an auxiliary problem Set x ðjÞ :¼ x if x is optimal in the problem above; otherwise let x ðjÞ be any vector satisfying the following conditions: x ðjÞ 2 X ðkþ1Þ ; f ðx ðjÞ ; t j Þ<0: Set DJ Let x ðijÞ :¼ x if vector x is optimal in problem ðAuxð2ÞÞ, otherwise choose any vector x ðijÞ 2 X ðkþ1Þ such that @f T ðx ðijÞ ; t j Þ @t a i ðjÞ<0: Otherwise (if at least one of the sets DJ ðjÞ is not empty), we set ( and proceed to the next iteration. The algorithm is described.

Justification of the algorithm DIIPS
To simplify the presentation, first suppose that we are applying the algorithm DIIPS to a linear w.r.t. x SIP problem in the form (P) that satisfies Assumption 4.1.
It is evident that the algorithm should stop after a finite number of iterations. Suppose that the algorithm has stopped on the ðk þ 1Þ-st iteration. Then, we have the sets J Since the function f(x, t) is supposed to be linear w.r.t. x and the set X ðkþ1Þ is convex, then there existsx 2 X ðkþ1Þ such that It follows from the algorithm DIIPS that f t j ; j 2 J ðkÞ Ã g & T Ã and qð t j ; lÞ>0 for Hence from Assumption 4.1, it follows Lemma 4.4. Let Assumption 4.1 be fulfilled and j 2 J ðkÞ Ã . Then there exists x Ãj 2 X such that 0j Þ: Proof. Define the following function: According to (16), we have qð t j ; lÞ ¼ 1; for all l ¼ Ab 6 ¼ 0;b 2 B: Then, by Definition 2.3, for each w ¼ 1; :::; n þ 1; there exists x ðwÞ 2 X; satisfying the following inequality: From the condition x ðwÞ 2 X, it follows that Fðx ðwÞ ;b r Þ 0 8r 6 ¼ w; r ¼ 1; :::; n þ 1: , where x Ãj 2 X; j 2 J ðkÞ Ã ; are the vectors considered in Lemma 4.4. Then,x Ã satisfies the following conditions: Moreover, we know that given an immobile index t j ; j 2 J Ã ¼ J ðkÞ Ã , for any x 2 X, it holds Then, evidently, for the constructed above vectorx Ã , we have Let z :¼ 1 2 ðx Ã þ xÞ 2 X, where x is the vector introduced in Section 4.1. Then, by construction, the following relations are satisfied: Given k 2 ½0; 1, let us consider vector xðkÞ :¼ ð1ÀkÞz þ kx; where vectorx 2 X ðkþ1Þ satisfies (15). Taking into account the linearity of f(x, t) w.r.t. x, we have Then, we can conclude that for 0<k<1, it holds It is evident that for a sufficiently small k>0, we can guarantee that there exists eðkÞ ! 0 such that eðkÞ ! 0 as k ! 0 and where T e t ð Þ ¼ s 2 T : jjtÀsjj e f g : Suppose that j 2 J ðkÞ Ã . Then, any t 2 T eðkÞ ð t i Þ can be presented in the form t ¼ t j þ Dt j ; Dt j 2 Lð t j Þ; jjDt j jj eðkÞ and in a rather small neighbourhood of t the following asymptotic expansion holds: where a j ¼ ða i ðjÞ; i 2 IðjÞÞ ! 0. Notice here that if ða i ðjÞ; i 2 IðjÞ n I ðkÞ 0 ðjÞÞ 6 ¼ 0, then the first-order term in the expansion above is negative. If ða i ðjÞ; i 2 IðjÞ n I ðkÞ 0 ðjÞÞ ¼ 0, then this term vanishes and we get In this case, f ðxðkÞ; tÞ<0 when ðb j ; a ðkÞ 0j Þ 6 ¼ 0 (taking into account (21)), and f ðxðkÞ; tÞ ¼ f ðxðkÞ; t j Þ ¼ 0 when ðb j ; a ðkÞ 0j Þ ¼ 0: Then for sufficiently small k>0. Therefore, it is proved that for a sufficiently small k>0 the vectorx ¼ xðkÞ has the following properties: (P1)x 2 X, i.e.x is a feasible solution of problem (P) (it follows from (22), (23)). (P2) The following relations are valid: Recall that by construction, f t j ; j 2 J and the algorithm DIIPS is justified.
In the case of convex (w.r.t. x) constraint functions, the steps of the algorithm are the same. To justify the algorithm in this case, one can use the same scheme as above, taking into account Lemmas 1-4 from Kostyukova and Tchemisova (2015).
Lemma 4.5. In Assumption 4.1, condition (11) is equivalent to the following statement: for any immobile index t 2 T Ã , there exists x ¼ xð tÞ 2 X such that vector t satisfies the sufficient conditions of strict local maximum in the problem max f x; t ð Þ; s:t: t 2 T; and these conditions have the form Proof. Suppose that Assumption 4.1 is satisfied. It was proved above that there exists a vectorx that satisfies properties (P1) and (P2). Hence, for any t 2 T Ã we can choose the vector x ¼ xð tÞ ¼x. Now let us consider a situation when for some t 2 T Ã there exists a vector x 2 X satisfying (24). If suppose that condition (11) is not satisfied, then we get that there exists l 2 Lð tÞ; l 6 ¼ 0 such that qð t; lÞ>1: Then, from the definition of the immobility order it follows that @f T x; t ð Þ @t l ¼ 0; l T @ 2 f x; t ð Þ @t l ¼ 0 8x 2 X: These equalities with x ¼ x 2 X contradict (24), and the lemma is proved.

Remarks
It is evident that if the constraint function f(x, t) is linear w.r.t. x, then the corresponding auxiliary problems (Aux(1)) and (Aux (2)) are linear w.r.t. x. On the base of Lemmas 1-4 from Kostyukova and Tchemisova (2015), one can prove that in the case of a convex w.r.t.
On the iterations of the algorithm, we do not need to solve the auxiliary problems (Aux(1)) and (Aux (2)). We only check the optimality of the given feasible solution x in each of these problems, and in the case when the solution is not optimal, find a feasible solution with a better (smaller) value of the cost function. Notice that this better solution is needed only for the justification of the algorithm. Often the sets L ðkÞ j ; j 2 J ðkÞ Ã , are either empty or consist of a finite number of elements. In these cases, in each of two problems (Aux(1)) and (Aux (2)), there is a finite number of constraints. In the general case, when a set L ðkÞ j consists of an infinite number of elements (it can be the union of a finite number of polyhedrons), at least one of the corresponding problem (Aux(1)) or (Aux (2)) possesses an infinite number of constraints But we should notice that these constraints are simpler than the original constraints f ðx; tÞ 0 since the function g(x, l) is quadratic w.r.t. the index variable l, while the dependence of the function f(x, t) on the index variable t, as a rule, is more complex.
The case when f(x, t) depends linearly on t, is not of particular interest since then the original SIP problem (1) can be reduced to a convex programming problem (with a finite number of constraints).