4,695 research outputs found
An easy way to obtain strong duality results in linear, linear semidefinite and linear semi-infinite programming
In linear programming it is known that an appropriate non-homogeneous Farkas Lemma leads to a short proof of the strong duality results for a pair of primal and dual programs. By using a corresponding generalized Farkas lemma we give a similar proof of the strong duality results for semidefinite programs under constraint qualifications. The proof includes optimality conditions. The same approach leads to corresponding results for linear semi-infinite programs. For completeness, the proofs for linear programs and the proofs of all auxiliary lemmata for the semidefinite case are included
Exact duality in semidefinite programming based on elementary reformulations
In semidefinite programming (SDP), unlike in linear programming, Farkas'
lemma may fail to prove infeasibility. Here we obtain an exact, short
certificate of infeasibility in SDP by an elementary approach: we reformulate
any semidefinite system of the form Ai*X = bi (i=1,...,m) (P) X >= 0 using only
elementary row operations, and rotations. When (P) is infeasible, the
reformulated system is trivially infeasible. When (P) is feasible, the
reformulated system has strong duality with its Lagrange dual for all objective
functions.
As a corollary, we obtain algorithms to generate the constraints of {\em all}
infeasible SDPs and the constraints of {\em all} feasible SDPs with a fixed
rank maximal solution.Comment: To appear, SIAM Journal on Optimizatio
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
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
Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization
Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes
Strong duality in conic linear programming: facial reduction and extended duals
The facial reduction algorithm of Borwein and Wolkowicz and the extended dual
of Ramana provide a strong dual for the conic linear program in the absence of any constraint qualification. The facial
reduction algorithm solves a sequence of auxiliary optimization problems to
obtain such a dual. Ramana's dual is applicable when (P) is a semidefinite
program (SDP) and is an explicit SDP itself. Ramana, Tuncel, and Wolkowicz
showed that these approaches are closely related; in particular, they proved
the correctness of Ramana's dual using certificates from a facial reduction
algorithm.
Here we give a clear and self-contained exposition of facial reduction, of
extended duals, and generalize Ramana's dual:
-- we state a simple facial reduction algorithm and prove its correctness;
and
-- building on this algorithm we construct a family of extended duals when
is a {\em nice} cone. This class of cones includes the semidefinite cone
and other important cones.Comment: A previous version of this paper appeared as "A simple derivation of
a facial reduction algorithm and extended dual systems", technical report,
Columbia University, 2000, available from
http://www.unc.edu/~pataki/papers/fr.pdf Jonfest, a conference in honor of
Jonathan Borwein's 60th birthday, 201
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