11 research outputs found
Generalized problem of linear copositive programming
Π‘ΡΠ°ΡΡΡ ΠΏΠΎΡΠ²ΡΡΠ΅Π½Π° ΠΈΠ·ΡΡΠ΅Π½ΠΈΡ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΎΠ½Π½ΡΡ
Π·Π°Π΄Π°Ρ, Π² ΠΊΠΎΡΠΎΡΡΡ
ΡΠ΅Π»Π΅Π²Π°Ρ ΡΡΠ½ΠΊΡΠΈΡ Π»ΠΈΠ½Π΅ΠΉΠ½Π° ΠΏΠΎ ΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠΌΠ΅ΡΠ½ΠΎΠΉ ΠΏΠ΅ΡΠ΅ΠΌΠ΅Π½Π½ΠΎΠΉ Ρ
, Π² ΡΠΎ Π²ΡΠ΅ΠΌΡ ΠΊΠ°ΠΊ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡ Π»ΠΈΠ½Π΅ΠΉΠ½Ρ ΠΏΠΎ Ρ
ΠΈ ΠΊΠ²Π°Π΄ΡΠ°ΡΠΈΡΠ½Ρ ΠΏΠΎ ΠΈΠ½Π΄Π΅ΠΊΡΡ t, ΠΏΡΠΈΠ½Π°Π΄Π»Π΅ΠΆΠ°ΡΠ΅ΠΌΡ Π·Π°Π΄Π°Π½Π½ΠΎΠΌΡ ΠΊΠΎΠ½ΡΡΡ. ΠΠ°Π΄Π°ΡΠΈ ΡΠ°ΠΊΠΎΠ³ΠΎ Π²ΠΈΠ΄Π° ΠΌΠΎΠ³ΡΡ ΠΈΠ½ΡΠ΅ΡΠΏΡΠ΅ΡΠΈΡΠΎΠ²Π°ΡΡΡΡ ΠΊΠ°ΠΊ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½ΠΈΠ΅ Π·Π°Π΄Π°Ρ ΠΏΠΎΠ»ΡΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΠΎΠ³ΠΎ
ΠΈ ΠΊΠΎΠΏΠΎΠ»ΠΎΠΆΠΈΡΠ΅Π»ΡΠ½ΠΎΠ³ΠΎ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠΈΡΠΎΠ²Π°Π½ΠΈΡ. ΠΠ»Ρ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΠΌΠΎΠΉ Π·Π°Π΄Π°ΡΠΈ ΡΠΎΡΠΌΡΠ»ΠΈΡΡΠ΅ΡΡΡ ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½Π°Ρ Π·Π°Π΄Π°ΡΠ° ΠΏΠΎΠ»ΡΠ±Π΅ΡΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠ³ΠΎ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΠΈ Π²Π²ΠΎΠ΄ΠΈΡΡΡ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²ΠΎ Π½Π΅ΠΏΠΎΠ΄Π²ΠΈΠΆΠ½ΡΡ
ΠΈΠ½Π΄Π΅ΠΊΡΠΎΠ², ΠΊΠΎΡΠΎΡΠΎΠ΅ Π»ΠΈΠ±ΠΎ ΠΏΡΡΡΠΎ, Π»ΠΈΠ±ΠΎ ΡΠ²Π»ΡΠ΅ΡΡΡ
ΠΎΠ±ΡΠ΅Π΄ΠΈΠ½Π΅Π½ΠΈΠ΅ΠΌ ΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠ³ΠΎ ΡΠΈΡΠ»Π° Π²ΡΠΏΡΠΊΠ»ΡΡ
ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½Π½ΡΡ
ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΠΈΠΊΠΎΠ². ΠΠ·ΡΡΠ΅Π½ΠΈΠ΅ ΡΠ²ΠΎΠΉΡΡΠ² ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π° Π΄ΠΎΠΏΡΡΡΠΈΠΌΡΡ
ΠΏΠ»Π°Π½ΠΎΠ² ΠΏΠΎΠ·Π²ΠΎΠ»ΠΈΠ»ΠΎ ΡΡΠΎΡΠΌΡΠ»ΠΈΡΠΎΠ²Π°ΡΡ ΠΈ Π΄ΠΎΠΊΠ°Π·Π°ΡΡ Π½ΠΎΠ²ΡΠ΅ ΡΡΡΠ΅ΠΊΡΠΈΠ²Π½ΡΠ΅ ΡΡΠ»ΠΎΠ²ΠΈΡ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΡΡΠΈ, ΠΊΠΎΡΠΎΡΡΠ΅ Π½Π΅ ΡΡΠ΅Π±ΡΡΡ
Π΄ΠΎΠΏΠΎΠ»Π½ΠΈΡΠ΅Π»ΡΠ½ΡΡ
ΡΡΠ»ΠΎΠ²ΠΈΠΉ Π½Π° ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡ ΠΈ ΠΈΠΌΠ΅ΡΡ ΡΠΎΡΠΌΡ ΠΊΡΠΈΡΠ΅ΡΠΈΠ΅Π².We consider a special class of optimization problems where the objective function is linear w.r.t. decision
variable Ρ
and the constraints are linear w.r.t. Ρ
and quadratic w.r.t. index t defined in a given cone. The problems of this class
can be considered as a generalization of semi-definite and copositive programming problems. For these problems, we
formulate an equivalent semi-infinite problem and define a set of immobile indices that is either empty or a union of a finite
number of convex bounded polyhedra. We have studied properties of the feasible sets of the problems under consideration and
use them to obtain new efficient optimality conditions for generalized copositive problems. These conditions are CQ-free and
have the form of criteria.publishe
ΠΠ±ΠΎΠ±ΡΠ΅Π½Π½Π°Ρ Π·Π°Π΄Π°ΡΠ° Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎΠ³ΠΎ ΠΊΠΎΠΏΠΎΠ»ΠΎΠΆΠΈΡΠ΅Π»ΡΠ½ΠΎΠ³ΠΎ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠΈΡΠΎΠ²Π°Π½ΠΈΡ
We consider a special class of optimization problems where the objective function is linear w.r.t. decision variable Ρ
and the constraints are linear w.r.t. Ρ
and quadratic w.r.t. index t defined in a given cone. The problems of this class can be considered as a generalization of semi-definite and copositive programming problems. For these problems, we formulate an equivalent semi-infinite problem and define a set of immobile indices that is either empty or a union of a finite number of convex bounded polyhedra. We have studied properties of the feasible sets of the problems under consideration and use them to obtain new efficient optimality conditions for generalized copositive problems. These conditions are CQ-free and have the form of criteria.Π‘ΡΠ°ΡΡΡ ΠΏΠΎΡΠ²ΡΡΠ΅Π½Π° ΠΈΠ·ΡΡΠ΅Π½ΠΈΡ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΎΠ½Π½ΡΡ
Π·Π°Π΄Π°Ρ, Π² ΠΊΠΎΡΠΎΡΡΡ
ΡΠ΅Π»Π΅Π²Π°Ρ ΡΡΠ½ΠΊΡΠΈΡ Π»ΠΈΠ½Π΅ΠΉΠ½Π° ΠΏΠΎ ΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠΌΠ΅ΡΠ½ΠΎΠΉ ΠΏΠ΅ΡΠ΅ΠΌΠ΅Π½Π½ΠΎΠΉ Ρ
, Π² ΡΠΎ Π²ΡΠ΅ΠΌΡ ΠΊΠ°ΠΊ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡ Π»ΠΈΠ½Π΅ΠΉΠ½Ρ ΠΏΠΎ Ρ
ΠΈ ΠΊΠ²Π°Π΄ΡΠ°ΡΠΈΡΠ½Ρ ΠΏΠΎ ΠΈΠ½Π΄Π΅ΠΊΡΡ t, ΠΏΡΠΈΠ½Π°Π΄Π»Π΅ΠΆΠ°ΡΠ΅ΠΌΡ Π·Π°Π΄Π°Π½Π½ΠΎΠΌΡ ΠΊΠΎΠ½ΡΡΡ. ΠΠ°Π΄Π°ΡΠΈ ΡΠ°ΠΊΠΎΠ³ΠΎ Π²ΠΈΠ΄Π° ΠΌΠΎΠ³ΡΡ ΠΈΠ½ΡΠ΅ΡΠΏΡΠ΅ΡΠΈΡΠΎΠ²Π°ΡΡΡΡ ΠΊΠ°ΠΊ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½ΠΈΠ΅ Π·Π°Π΄Π°Ρ ΠΏΠΎΠ»ΡΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΠΎΠ³ΠΎ ΠΈ ΠΊΠΎΠΏΠΎΠ»ΠΎΠΆΠΈΡΠ΅Π»ΡΠ½ΠΎΠ³ΠΎ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠΈΡΠΎΠ²Π°Π½ΠΈΡ. ΠΠ»Ρ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΠΌΠΎΠΉ Π·Π°Π΄Π°ΡΠΈ ΡΠΎΡΠΌΡΠ»ΠΈΡΡΠ΅ΡΡΡ ΡΠΊΠ²ΠΈΠ²Π°Π»Π΅Π½ΡΠ½Π°Ρ Π·Π°Π΄Π°ΡΠ° ΠΏΠΎΠ»ΡΠ±Π΅ΡΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠ³ΠΎ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΠΈ Π²Π²ΠΎΠ΄ΠΈΡΡΡ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²ΠΎ Π½Π΅ΠΏΠΎΠ΄Π²ΠΈΠΆΠ½ΡΡ
ΠΈΠ½Π΄Π΅ΠΊΡΠΎΠ², ΠΊΠΎΡΠΎΡΠΎΠ΅ Π»ΠΈΠ±ΠΎ ΠΏΡΡΡΠΎ, Π»ΠΈΠ±ΠΎ ΡΠ²Π»ΡΠ΅ΡΡΡ ΠΎΠ±ΡΠ΅Π΄ΠΈΠ½Π΅Π½ΠΈΠ΅ΠΌ ΠΊΠΎΠ½Π΅ΡΠ½ΠΎΠ³ΠΎ ΡΠΈΡΠ»Π° Π²ΡΠΏΡΠΊΠ»ΡΡ
ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½Π½ΡΡ
ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΠΈΠΊΠΎΠ². ΠΠ·ΡΡΠ΅Π½ΠΈΠ΅ ΡΠ²ΠΎΠΉΡΡΠ² ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π° Π΄ΠΎΠΏΡΡΡΠΈΠΌΡΡ
ΠΏΠ»Π°Π½ΠΎΠ² ΠΏΠΎΠ·Π²ΠΎΠ»ΠΈΠ»ΠΎ ΡΡΠΎΡΠΌΡΠ»ΠΈΡΠΎΠ²Π°ΡΡ ΠΈ Π΄ΠΎΠΊΠ°Π·Π°ΡΡ Π½ΠΎΠ²ΡΠ΅ ΡΡΡΠ΅ΠΊΡΠΈΠ²Π½ΡΠ΅ ΡΡΠ»ΠΎΠ²ΠΈΡ ΠΎΠΏΡΠΈΠΌΠ°Π»ΡΠ½ΠΎΡΡΠΈ, ΠΊΠΎΡΠΎΡΡΠ΅ Π½Π΅ ΡΡΠ΅Π±ΡΡΡ Π΄ΠΎΠΏΠΎΠ»Π½ΠΈΡΠ΅Π»ΡΠ½ΡΡ
ΡΡΠ»ΠΎΠ²ΠΈΠΉ Π½Π° ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡ ΠΈ ΠΈΠΌΠ΅ΡΡ ΡΠΎΡΠΌΡ ΠΊΡΠΈΡΠ΅ΡΠΈΠ΅Π²
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 (orK-semidefinite) programming problems, where the setKis a polyhedral convex cone. For these problems, we introduce theconcept of immobile indices and study the properties of the set of normalized immobile indices and the feasible set. Thisstudy provides the main result of the paper, which is to formulate and prove the new first-order optimality conditions inthe form of a criterion. The optimality conditions are explicit and do not use any constraint qualifications. For the case of alinear cost function, we reformulate theK-semidefinite problem in a regularized form and construct its dual. We show thatthe pair of the primal and dual regularized problems satisfies the strong duality relation which means that the duality gap is vanishing.publishe
Optimality conditions for linear copositive programming problems with isolated immobile indices
In the present paper, we apply our recent results on optimality for convex semi-infinite programming to a problem of linear copositive programming (LCP). 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 LCP. 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.publishe
Immobile indices and CQ-free optimality criteria for linear copositive programming problems
We consider problems of linear copositive programming where feasible sets consist of vectors
for which the quadratic forms induced by the corresponding linear matrix combinations
are nonnegative over the nonnegative orthant. Given a linear copositive problem, we define
immobile indices of its constraints and a normalized immobile index set. We prove that the
normalized immobile index set is either empty or can be represented as a union of a finite
number of convex closed bounded polyhedra. We show that the study of the structure of
this set and the connected properties of the feasible set permits to obtain new optimality
criteria for copositive problems. These criteria do not require the fulfillment of any additional
conditions (constraint qualifications or other). An illustrative example shows that the
optimality conditions formulated in the paper permit to detect the optimality of feasible
solutions for which the known sufficient optimality conditions are not able to do this. We
apply the approach based on the notion of immobile indices to obtain new formulations of
regularized primal and dual problems which are explicit and guarantee strong duality.publishe
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 have similar structure and
properties as that proposed in the works of M. Ramana, L. Tuncel, and H. Wolkovicz, for
semidefinite programming, but are obtained using different techniques.publishe
Conic Programming Approaches for Polynomial Optimization: Theory and Applications
Historically, polynomials are among the most popular class of functions used for empirical modeling in science and engineering. Polynomials are easy to evaluate, appear naturally in many physical (real-world) systems, and can be used to accurately approximate any smooth function. It is not surprising then, that the task of solving polynomial optimization problems; that is, problems where both the objective function and constraints are multivariate polynomials, is ubiquitous and of enormous interest in these fields. Clearly, polynomial op- timization problems encompass a very general class of non-convex optimization problems, including key combinatorial optimization problems.The focus of the first three chapters of this document is to address the solution of polynomial optimization problems in theory and in practice, using a conic optimization approach. Convex optimization has been well studied to solve quadratic constrained quadratic problems. In the first part, convex relaxations for general polynomial optimization problems are discussed. Instead of using the matrix space to study quadratic programs, we study the convex relaxations for POPs through a lifted tensor space, more specifically, using the completely positive tensor cone and the completely positive semidefinite tensor cone. We show that tensor relaxations theoretically yield no-worse global bounds for a class of polynomial optimization problems than relaxation for a QCQP reformulation of the POPs. We also propose an approximation strategy for tensor cones and show empirically the advantage of the tensor relaxation.In the second part, we propose an alternative SDP and SOCP hierarchy to obtain global bounds for general polynomial optimization problems. Comparing with other existing SDP and SOCP hierarchies that uses higher degree sum of square (SOS) polynomials and scaled diagonally sum of square polynomials (SDSOS) when the hierarchy level increases, these proposed hierarchies, using fixed degree SOS and SDSOS polynomials but more of these polynomials, perform numerically better. Numerical results show that the hierarchies we proposed have better performance in terms of tightness of the bound and solution time compared with other hierarchies in the literature.The third chapter deals with Alternating Current Optimal Power Flow problem via a polynomial optimization approach. The Alternating Current Optimal Power Flow (ACOPF) problem is a challenging non-convex optimization problem in power systems. Prior research mainly focuses on using SDP relaxations and SDP-based hierarchies to address the solution of ACOPF problem. In this Chapter, we apply existing SOCP hierarchies to this problem and explore the structure of the network to propose simplified hierarchies for ACOPF problems. Compared with SDP approaches, SOCP approaches are easier to solve and can be used to approximate large scale ACOPF problems.The last chapter also relates to the use of conic optimization techniques, but in this case to pricing in markets with non-convexities. Indeed, it is an application of conic optimization approach to solve a pricing problem in energy systems. Prior research in energy market pricing mainly focus on linear costs in the objective function. Due to the penetration of renewable energies into the current electricity grid, it is important to consider quadratic costs in the objective function, which reflects the ramping costs for traditional generators. This study address the issue how to find the market clearing prices when considering quadratic costs in the objective function
Copositive programming via semi-infinite optimization
Copositive programming (CP) can be regarded as a special instance of linear semi-infinite programming (SIP). We study CP from the viewpoint of SIP and discuss optimality and duality results. Different approximation schemes for solving CP are interpreted as discretization schemes in SIP. This leads to sharp explicit error bounds for the values and solutions in dependence on the mesh size. Examples illustrate the structure of the original program and the approximation schemes