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Numerical solution of an obstacle problem with interval coefficients

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  • In this work we propose a novel numerical method for a finite-dimensional optimization problem arising from the discretization of an infinite-dimensional constrained optimization problem, called an obstacle problem, with interval coefficients. In this method, the two different ways of characterizing the optimal solutions, i.e., minimizing the mid-point and one end-point (the worst-case scenario) or the mid-point and the width of the objective interval, are formulated as a single constrained multi-objective minimization problem and the KKT conditions of the optimization problem defining the Pareto optimal solution to the multi-objective problem are of the form of a Linear Complementarity Problem (LCP) which is shown to have a unique solution. The LCP is the approximated by a non-linear equation using an interior penalty approach. We prove that the penalty equation is uniquely solvable and its solution converges to that of LCP as the penalty constant approaches to zero. Numerical results are presented to demonstrate the usefulness of the numerical method proposed.

    Mathematics Subject Classification: Primary: 65K15, 90C70; Secondary: 65G99, 90C29.

    Citation:

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  • Figure 1.  Computed optimal solution of Test 1 when $ \theta = 0 $

    Figure 2.  Computed optimal solution of Test 1 when $ \theta = 0.5 $

    Figure 3.  Computed optimal solution of Test 2 when $ \theta = 0 $

    Figure 4.  Computed optimal solution of Test 2 when $ \theta = 0.5 $

    Figure 5.  Computed solution at $ \lambda = 0.5 $; (a) and solution at $ \theta = 0 $, (b) difference between solutions of $ \theta = 0 $ and 0.5

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