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The Smoothing Function Method For A Class Of Inverse Second Order Cone Programming Problems

Posted on:2009-10-07Degree:MasterType:Thesis
Country:ChinaCandidate:Y TianFull Text:PDF
GTID:2120360242484955Subject:Operational Research and Cybernetics
Abstract/Summary:PDF Full Text Request
Second-Order-Cone Programming (SOCP) problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of second-order-cones. Linear programs, convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as SOCP problems. These problems model applications from a broad range of fields from engineering, control and finance to robust optimization and combinatorial optimization. Although SOCP problems can be solved as semidefmite programming problems, this way is not suggested for the losing of some good properties of second-order-cone. Thus there are many important contributions to SOCP problems basing on differential properties of the second order cones. But for inverse SOCP problems, few works are known. In an optimization model, usually there are parameters associated with decision variables in the objective function or in the constraint set. When solving the optimization problem, it usually assumes that these parameter values are known and it needs to find an optimal solution to the given model. However, there are many instances in practice, in which it only has some estimates for parameter values, but may know certain optimal solutions from experience, observations or experiments. An inverse optimization problem is to find values of parameters which make the known solutions optimal and which differ from the given estimates as little as possible.The thesis considers an inverse linear programming problem with second-order-cone constraints in which the parameters in the constraint set need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. With the use of KKT system, we formulate this problem as a nonlinear complementarity constrained nonsmooth problem in which the objective function is quadratic. With the help of smoothing function, a family of smoothing problems is used to approximate the original problem and the convergence results are demonstrated. Finally multiplier method based on augmented Lagrange function is used to solve the smoothing problem. And the quasi-Newton method with Wolfe linear search rule is used to solve the unconstrained subproblem. Furthermore numerical experiments by the algorithm are reported, which indicate that the algorithm is effective for solving this class of inverse second-order-cone programming problems. The study on the proposed smoothing function and the multiplier method involves the theoretic results from Jordan algebra and differential properties of the smoothing function, respectively. All of these issues have been discussed in this paper.
Keywords/Search Tags:Second Order Cone Programming, Inverse Problem, Smoothing Function Method, Multiply method, Quasi-Newton Method based on BFGS Correction
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