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LARGESCALE LINEARLY CONSTRAINED OPTIMIZATION
, 1978
"... An algorithm for solving largescale nonlinear ' programs with linear constraints is presented. The method combines efficient sparsematrix techniques as in the revised simplex method with stable quasiNewton methods for handling the nonlinearities. A generalpurpose production code (MINOS) is ..."
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Cited by 100 (17 self)
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An algorithm for solving largescale nonlinear ' programs with linear constraints is presented. The method combines efficient sparsematrix techniques as in the revised simplex method with stable quasiNewton methods for handling the nonlinearities. A generalpurpose production code (MINOS) is described, along with computational experience on a wide variety of problems.
Semidefinite Programming for Assignment and Partitioning Problems
, 1996
"... Semidefinite programming, SDP, is an extension of linear programming, LP, where the nonnegativity constraints are replaced by positive semidefiniteness constraints on matrix variables. SDP has proven successful in obtaining tight relaxations for NP hard combinatorial optimization problems of simpl ..."
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Cited by 14 (2 self)
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Semidefinite programming, SDP, is an extension of linear programming, LP, where the nonnegativity constraints are replaced by positive semidefiniteness constraints on matrix variables. SDP has proven successful in obtaining tight relaxations for NP hard combinatorial optimization problems of simple structure such as the maxcut and graph bisection problems. In this work, we try to solve more complicated combinatorial problems such as the quadratic assignment, general graph partitioning and set partitioning problems. A tight SDP relaxation can be obtained by exploiting the geometrical structure of the convex hull of the feasible points of the original combinatorial problem. The analysis of the structure enables us to find the socalled "minimal face" and "gangster operator" of the SDP. This plays a significant role in simplifying the problem and enables us to derive a unified SDP relaxation for the three different problems. We develop an efficient "partial infeasible" primaldual inter...
SOLVING COMBINATORIAL OPTIMISATION PROBLEMS USING NEURAL NETWORKS
, 1996
"... Combinatorial optimisation problems (COP's) arise naturally when mathematically modelling many practical optimisation problems from science and engineering. Due to the NPhard nature of many COP's, heuristics are often used to provide rapid and nearoptimal solutions. Neural networks are a ..."
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Cited by 1 (0 self)
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Combinatorial optimisation problems (COP's) arise naturally when mathematically modelling many practical optimisation problems from science and engineering. Due to the NPhard nature of many COP's, heuristics are often used to provide rapid and nearoptimal solutions. Neural networks are a novel and potentially powerful alternative approach to solving such problems. They are also intrinsically parallel, with much potential for rapid hardware implementation. Unfortunately, existing neural techniques are widely considered to be unsuited to optimisation due to their tendency to produce infeasible or poor quality solutions. Over the last decade or so, two main types of neural networks have been proposed for solving COP's in particular, the Travelling Salesman Problem (TSP). The first of these neural approaches is the Hopfield neural network which evolves in such away asto minimise a system energy function. In its original form, the Hopfield energy function involves many parameters which need to be tuned, and constructing a suitable energy function which enables the network to arrive at feasible nearoptimal solutions is a difficult
Semidefinite Programming Relaxations For Set Partitioning Problems
, 1996
"... We present a relaxation for the set partitioning problem that combines the standard linear programming relaxation with a semidefinite programming relaxation. We include numerical results that illustrate the strength and efficiency of this relaxation. Contents 1 INTRODUCTION 2 1.1 Background . . . ..."
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We present a relaxation for the set partitioning problem that combines the standard linear programming relaxation with a semidefinite programming relaxation. We include numerical results that illustrate the strength and efficiency of this relaxation. Contents 1 INTRODUCTION 2 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 SDP RELAXATION 3 3 NUMERICAL TESTS 8 4 SDP RELAXATION FOR LARGE SPARSE PROBLEMS 8 4.1 An SDP Relaxation with Block Structure . . . . . . . . . . . . . . . . . . . . . . 8 4.2 An Infeasible PrimalDual InteriorPoint Method . . . . . . . . . . . . . . . . . . 13 4.3 Preliminary Numerical Tests and Future Work . . . . . . . . . . . . . . . . . . . 15 A APPENDIXNotation 16 This report is available by anonymous ftp at orion.uwaterloo.ca in the directory pub/henry/reports; or over WWW with URL ftp://orion.uwaterloo.ca/pub/henry/reports/ABSTRACTS.html y University of Waterloo, Department of Combinatorics and Optim...