Considering copositivity locally

Dickinson, Peter J. C.; Hildebrand, Roland

doi:10.1016/j.jmaa.2016.01.063

Cited by 26 publications

(17 citation statements)

References 16 publications

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“…Different algorithms for copositivity detection are described e.g. in [12,[14][15][16]. A clustered bibliography on copositive optimization can be found in [17].…”

Section: Introductionmentioning

confidence: 99%

Optimality conditions for linear copositive programming problems with isolated immobile indices

Kostyukova

Tchemisova

2018

Optimization

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In the present paper, we apply our recent results on optimality for convex Semi-Infinite Programming to a problem of Linear Copositive Programming. We prove explicit optimality conditions that use concepts of immobile indices and their immobility orders, and do not require the Slater constraint qualification to be satisfied. The only assumption that we impose here is that the set of immobile indices consists of isolated points, and hence is finite. This assumption is weaker than the Slater condition; therefore, the optimality conditions obtained in the paper are more general when compared with those usually used in Linear Copositive Programming. We present an example of a problem in which the new optimality conditions allow one to test the optimality of a given feasible solution while the known optimality conditions fail to do this. Further, we use the immobile indices to construct a pair of regularized dual copositive problems and show that regardless of whether the Slater condition is satisfied or not, the duality gap between the optimal values of these problems is zero. An example of a problem is presented for which the standard strict duality fails, but the duality gap obtained by using the regularized dual problem vanishes.

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“…Different algorithms for copositivity detection are described e.g. in [12,[14][15][16]. A clustered bibliography on copositive optimization can be found in [17].…”

Section: Introductionmentioning

confidence: 99%

Optimality conditions for linear copositive programming problems with isolated immobile indices

Kostyukova

Tchemisova

2018

Optimization

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“…It worth mentioning that several attempts have been done to study the facial structure of the cones of copositive and completely positive matrices. Thus in [3,9,11], the authors give explicit characterizations of extreme rays (faces of dimension one) of copositive cones of dimensions five and six. In [8,9], some properties of special types of faces (minimal and maximal ones) are studied.…”

Section: Introductionmentioning

confidence: 99%

“…Thus in [3,9,11], the authors give explicit characterizations of extreme rays (faces of dimension one) of copositive cones of dimensions five and six. In [8,9], some properties of special types of faces (minimal and maximal ones) are studied. Nevertheless, until now, for the cone of copositive matrices, faces of this cone and the corresponding dual cones are not well studied.…”

Section: Introductionmentioning

confidence: 99%

On equivalent representations and properties of faces of the cone of copositive matrices

Kostyukova

Tchemisova

2022

Optimization

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The paper is devoted to a study of the cone COP p of copositive matrices. Based on the known from semi-infinite optimization concept of immobile indices, we define zero and minimal zero vectors of a subset of the cone COP p and use them to obtain different representations of faces of COP p and the corresponding dual cones. We describe the minimal face of COP p containing a given convex subset of this cone and prove some propositions that allow to obtain equivalent descriptions of the feasible sets of a copositive problems and may be useful for creating new numerical methods based on their regularization.

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“…A fruitful concept in the study of copositive matrices is that of zeros and their supports, initiated in the works of Baumert [2], [3], see also [16], [7], and [19] for further developments and applications. A non-zero vector u ∈ R n + is called a zero of a copositive matrix A ∈ C n if u T Au = 0.…”

Section: Introductionmentioning

confidence: 99%

Copositive matrices with circulant zero support set

Hildebrand¹

2017

Linear Algebra and its Applications

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International audienceLet n≥5 and let u_1,…,u_n be nonnegative real n-vectors such that the indices of their positive elements form the sets {1,2,…,n−2},{2,3,…,n−1},…,{n,1,…,n−3}, respectively. Here each index set is obtained from the previous one by a circular shift. The set of copositive forms which vanish on the vectors u_1,…,u_n is a face of the copositive cone C^n. We give an explicit semi-definite description of this face and of its subface consisting of positive semi-definite forms, and study their properties. If the vectors u_1,…,u_n and their positive multiples exhaust the zero set of an exceptional copositive form belonging to this face, then we say it has minimal circulant zero support set, and otherwise non-minimal circulant zero support set. We show that forms with non-minimal circulant zero support set are always extremal, and forms with minimal circulant zero support sets can be extremal only if n is odd. We construct explicit examples of extremal forms with non-minimal circulant zero support set for any order n≥5, and examples of extremal forms with minimal circulant zero support set for any odd order n≥5n≥5. The set of all forms with non-minimal circulant zero support set, i.e., defined by different collections u_1,…,u_n of zeros, is a submanifold of codimension 2n, the set of all forms with minimal circulant zero support set a submanifold of codimension n

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Considering copositivity locally

Cited by 26 publications

References 16 publications

Optimality conditions for linear copositive programming problems with isolated immobile indices

Optimality conditions for linear copositive programming problems with isolated immobile indices

On equivalent representations and properties of faces of the cone of copositive matrices

Copositive matrices with circulant zero support set

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