A Computationally Fast and Approximate Method for Karush-Kuhn-Tucker Proximity Measure

Karush-Kuhn-Tucker (KKT) optimality conditions are used for checking whether a solution obtained by an optimization algorithm is truly an optimal solution or not by theoretical and applied optimization researchers. When a point is not the true optimum, a simple violation measure of KKT optimality conditions cannot indicate anything about it’s proximity to the optimal solution. Past few studies by the first author and his collaborators suggested a KKT proximity measure (KKTPM) that is able to identify relative closeness of any point from the theoretical optimum point without actually knowing the exact location of the optimum point. In this paper, we suggest several computationally fast methods for computing an approximate KKTPM value, so that a convergence measure for iteration-wise best solutions of an optimization algorithm can be quantified for terminating an optimization run. The KKTPM value can also be used to isolate less-converged solutions in a population-based optimization algorithm so as to specially modify them for an overall faster execution of the optimization run. The approximate KKTPM values are evaluated in comparison with the original exact KKTPM value on standard single-objective, multi-objective and many-objective optimization problems. In all cases, our proposed ‘estimated’ approximate method is found to achieve a strong correlation of KKTPM values with the exact values and achieve such results in two or three orders of magnitude smaller computational time. These results are extremely motivating to launch further studies in using the proposed estimated KKTPM procedure for establishing termination-based and developing other modified optimization procedures.

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