198 research outputs found
Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets
In the present paper, we analyze a class of convex Semi-Infinite Programming
problems with arbitrary index sets defined by a finite number of nonlinear inequalities. The analysis is carried out by employing the constructive approach, which, in turn, relies on the notions of immobile indices and their immobility orders. Our previous work showcasing this approach includes a number of papers dealing with simpler cases of semi-infinite problems than the ones under consideration here. Key findings of the paper include the formulation and the proof of implicit and explicit optimality conditions under assumptions, which are less restrictive than the constraint qualifications traditionally used. In this perspective, the optimality conditions in question are also compared to those provided in the relevant literature.
Finally, the way to formulate the obtained optimality conditions is demonstrated by applying the results of the paper to some special cases of the convex semi-infinite problem
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
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
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
SOS-convex Semi-algebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization
In this paper, we introduce a new class of nonsmooth convex functions called
SOS-convex semialgebraic functions extending the recently proposed notion of
SOS-convex polynomials. This class of nonsmooth convex functions covers many
common nonsmooth functions arising in the applications such as the Euclidean
norm, the maximum eigenvalue function and the least squares functions with
-regularization or elastic net regularization used in statistics and
compressed sensing. We show that, under commonly used strict feasibility
conditions, the optimal value and an optimal solution of SOS-convex
semi-algebraic programs can be found by solving a single semi-definite
programming problem (SDP). We achieve the results by using tools from
semi-algebraic geometry, convex-concave minimax theorem and a recently
established Jensen inequality type result for SOS-convex polynomials. As an
application, we outline how the derived results can be applied to show that
robust SOS-convex optimization problems under restricted spectrahedron data
uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP
relaxation result for restricted ellipsoidal data uncertainty and answers the
open questions left in [Optimization Letters 9, 1-18(2015)] on how to recover a
robust solution from the semi-definite programming relaxation in this broader
setting
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