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A New Relaxation Framework for Quadratic Assignment Problems based on Matrix Splitting
, 2010
"... Quadratic assignment problems (QAPs) are known to be among the hardest discrete optimization problems. Recent study shows that even obtaining a strong lower bound for QAPs is a computational challenge. In this paper, we first discuss how to construct new simple convex relaxations of QAPs based on va ..."
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Quadratic assignment problems (QAPs) are known to be among the hardest discrete optimization problems. Recent study shows that even obtaining a strong lower bound for QAPs is a computational challenge. In this paper, we first discuss how to construct new simple convex relaxations of QAPs based on various matrix splitting schemes. Then we introduce the socalled symmetric mappings that can be used to derive strong cuts for the proposed relaxation model. We show that the bounds based on the new models are comparable to some strong bounds in the literature. Promising experimental results based on the new relaxations will be reported.
VSDP: A MATLAB software package for verified semidefinite programming
 In Conference paper of NOLTA 2006, p. 327330, 2006. URL http://www.ti3.tuharburg. de/paper/jansson/Nolta06.pdf
"... Abstract — VSDP is a MATLAB software package for solving rigorously semidefinite programming problems. Functions for computing verified forward error bounds of the true optimal value and verified certificates of feasibility and infeasibility are provided. All rounding errors due to floating point ar ..."
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Abstract — VSDP is a MATLAB software package for solving rigorously semidefinite programming problems. Functions for computing verified forward error bounds of the true optimal value and verified certificates of feasibility and infeasibility are provided. All rounding errors due to floating point arithmetic are taken into account. 1.
VSDP: Verified SemiDefinite Programming USER’S GUIDE, Beta Version 0.1 for MATLAB 7.0
, 2006
"... VSDP is a MATLAB software package for rigorously solving semidefinite programming problems. It expresses these problems in a notation closely related to the form given in textbooks and scientific papers. Functions for computing verified forward error bounds of the true optimal value and verified cer ..."
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Cited by 3 (2 self)
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VSDP is a MATLAB software package for rigorously solving semidefinite programming problems. It expresses these problems in a notation closely related to the form given in textbooks and scientific papers. Functions for computing verified forward error bounds of the true optimal value and verified certificates of feasibility and infeasibility are provided. All rounding errors due to floating point arithmetic are taken into account. Computational results are given, including results for the SDPLIB benchmark problems. This package supports interval input data and sparse format. 1 Copyright (C) 2006 Christian Jansson
A generic ellipsoid abstract domain for linear time invariant systems
 In HSCC
, 2012
"... Abstract. Embedded system control often relies on linear systems, which admit quadratic invariants. The parts of the code that host linear system implementations need dedicated analysis tools since intervals or linear abstract domains will give imprecise results, if any at all, on these systems. Re ..."
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Abstract. Embedded system control often relies on linear systems, which admit quadratic invariants. The parts of the code that host linear system implementations need dedicated analysis tools since intervals or linear abstract domains will give imprecise results, if any at all, on these systems. Reference [9] proposes a specific abstraction for digital filters that addresses this issue on a specific class of controllers. This paper aims at generalizing the idea, relying on existing methods from Control Theory to automatically generate quadratic invariants for linear time invariant systems, whose stability is provable. This class encompasses nth order digital filters and, in general, controllers embedded in critical systems. While control theorists only focus on the existence of such invariants, this paper proposes a method to effectively compute tight ones. The method has been implemented and applied to some benchmark systems, giving good results. It also considers floating points issues and validates the soundness of the computed invariants.
A LagrangianDNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems
, 2013
"... We propose an efficient computational method for linearly constrained quadratic optimization problems (QOPs) with complementarity constraints based on their Lagrangian and doubly nonnegative (DNN) relaxation and firstorder algorithms. The simplified LagrangianCPP relaxation of such QOPs proposed b ..."
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We propose an efficient computational method for linearly constrained quadratic optimization problems (QOPs) with complementarity constraints based on their Lagrangian and doubly nonnegative (DNN) relaxation and firstorder algorithms. The simplified LagrangianCPP relaxation of such QOPs proposed by Arima, Kim, and Kojima in 2012 takes one of the simplest forms, an unconstrained conic linear optimization problem with a single Lagrangian parameter in a completely positive (CPP) matrix variable with its upperleft element fixed to 1. Replacing the CPP matrix variable by a DNN matrix variable, we derive the LagrangianDNN relaxation, and establish the equivalence between the optimal value of the DNN relaxation of the original QOP and that of the LagrangianDNN relaxation. We then propose an efficient numerical method for the LagrangianDNN relaxation using a bisection method combined with the proximal alternating direction multiplier and the accelerated proximal gradient methods. Numerical results on binary QOPs, quadratic multiple knapsack problems, maximum stable set problems, and quadratic assignment problems illustrate the superior performance of the proposed method for attaining tight lower bounds in shorter computational time.
Guaranteed Accuracy for Conic Programming Problems in Vector Lattices.
, 2007
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COUPLING POLICY ITERATION WITH SEMIDEFINITE RELAXATION TO COMPUTE ACCURATE NUMERICAL INVARIANTS IN STATIC ANALYSIS
"... We introduce a new domain for finding precise numerical invariants of programs by abstract interpretation. This domain, which consists of sublevel sets of nonlinear functions, generalizes the domain of linear templates introduced by Manna, Sankaranarayanan, and Sipma. In the case of quadratic te ..."
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We introduce a new domain for finding precise numerical invariants of programs by abstract interpretation. This domain, which consists of sublevel sets of nonlinear functions, generalizes the domain of linear templates introduced by Manna, Sankaranarayanan, and Sipma. In the case of quadratic templates, we use Shor’s semidefinite relaxation to derive computable yet precise abstractions of semantic functionals, and we show that the abstract fixpoint equation can be solved accurately by coupling policy iteration and semidefinite programming. We demonstrate the interest of our approach on a series of examples (filters, integration schemes) including a degenerate one (symplectic scheme).
IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1 On Computation of Performance Bounds of Optimal Index Assignment
"... Abstract—Channeloptimized index assignment of source codewords is arguably the simplest way of improving transmission error resilience, while keeping the source and/or channel codes intact. But optimal design of index assignment is an instance of quadratic assignment problem (QAP), one of the harde ..."
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Abstract—Channeloptimized index assignment of source codewords is arguably the simplest way of improving transmission error resilience, while keeping the source and/or channel codes intact. But optimal design of index assignment is an instance of quadratic assignment problem (QAP), one of the hardest optimization problems in the NPcomplete class. In this work we make a progress in the research of index assignment optimization. We apply some recent results of QAP research to compute the strongest lower bounds so far for channel distortion of BSC among all index assignments. The strength of the resulting lower bounds is validated by comparing them against the upper bounds produced by heuristic index assignment algorithms. Index Terms—Index assignment, quantization, error resilience, quadratic assignment, semidefinite programming, relaxation. I.