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EXACT DUALITY IN SEMIDEFINITE PROGRAMMING BASED ON ELEMENTARY REFORMULATIONS
, 2015
"... 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 equality constrained semidefinite system using only elementary row operatio ..."
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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 equality constrained semidefinite system using only elementary row operations, and rotations. When a system is infeasible, the reformulated system is trivially infeasible. When a system 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 all infeasible SDPs and the constraints of all feasible SDPs with a fixed rank maximal solution. Our elementary reformulations can be constructed either by a direct method, or by adapting the WakiMuramatsu facial reduction algorithm. In different language, the reformulations provide a standard form of spectrahedra, to easily verify either their emptiness, or a tight upper bound on the rank of feasible solutions.
CBLIB 2014: A benchmark library for conic mixedinteger and continuous optimization. Optimization Online
, 2014
"... Abstract The Conic Benchmark Library (CBLIB 2014) is a collection of more than a hundred conic optimization instances under a free and open license policy. It is the first extensive benchmark library for the advancing field of conic mixedinteger and continuous optimization, which is already suppor ..."
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Abstract The Conic Benchmark Library (CBLIB 2014) is a collection of more than a hundred conic optimization instances under a free and open license policy. It is the first extensive benchmark library for the advancing field of conic mixedinteger and continuous optimization, which is already supported by all major commercial solvers and spans a wide range of industrial applications. The library addresses the particular need for public test sets mixing cone types and allowing integer variables, but has all types of conic optimization in target. The CBF file format is embraced as standard, and tools are provided to aid integration with, or transformation to the input format of, any software package.
Weak Infeasibility in Second Order Cone Programming
"... Abstract The objective of this work is to study weak infeasibility in second order cone programming. For this purpose, we consider a relaxation sequence of feasibility problems that mostly preserve the feasibility status of the original problem. This is used to show that for a given weakly infeasib ..."
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Abstract The objective of this work is to study weak infeasibility in second order cone programming. For this purpose, we consider a relaxation sequence of feasibility problems that mostly preserve the feasibility status of the original problem. This is used to show that for a given weakly infeasible problem at most m directions are needed to approach the cone, where m is the number of Lorentz cones. We also tackle a closely related question and show that given a bounded optimization problem satisfying Slater's condition, we may transform it into another problem that has the same optimal value but it is ensured to attain it. From solutions to the new problem, we discuss how to obtain solution to the original problem which are arbitrarily close to optimality. Finally, we discuss how to obtain finite certificate of weak infeasibility by combining our own techniques with facial reduction. The analysis is similar in spirit to previous work by the authors on SDPs, but a different approach is required to obtain tighter bounds. Keywords: weak infeasibility second order cone programming feasibility problem.
Bad semidefinite programs: they all look the same
"... Duality theory plays a central role in semidefinite programming, since in optimization algorithms a dual solution serves as a certificate of optimality. However, in semidefinite duality pathological phenomena occur: nonattainment of the optimal value, positive duality gaps, and infeasibility of the ..."
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Duality theory plays a central role in semidefinite programming, since in optimization algorithms a dual solution serves as a certificate of optimality. However, in semidefinite duality pathological phenomena occur: nonattainment of the optimal value, positive duality gaps, and infeasibility of the dual, even when the primal is bounded. We say that the semidefinite system PSD = {x  ∑ m i=1 xiAi ≼ B} is badly behaved, or lacks uniform LPduality, if for some linear objective function c the value sup{〈c,x〉x ∈ PSD} is finite, but the dual program has no solution attaining the same value. We give simple, and exact characterizations of badly behaved semidefinite systems. Surprisingly, it turns out that the system α 1 1 0 x1 ≼ , (0.1)
Facially exposed cones are not always nice
"... Abstract We address the conjecture proposed by Gábor Pataki that every facially exposed cone is nice. We show that the conjecture is true in the threedimensional case, however, there exists a fourdimensional counterexample of a cone that is facially exposed but is not nice. ..."
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Abstract We address the conjecture proposed by Gábor Pataki that every facially exposed cone is nice. We show that the conjecture is true in the threedimensional case, however, there exists a fourdimensional counterexample of a cone that is facially exposed but is not nice.