A Sequential Quadratic Programming Method for Constrained Multi-objective Optimization Problems
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A Sequential Quadratic Programming Method for Constrained Multi-objective Optimization Problems Md Abu Talhamainuddin Ansary1
· Geetanjali Panda1
Received: 1 March 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this article, a globally convergent sequential quadratic programming (SQP) method is developed for multi-objective optimization problems with inequality type constraints. A feasible descent direction is obtained using a linear approximation of all objective functions as well as constraint functions. The sub-problem at every iteration of the sequence has feasible solution. A non-differentiable penalty function is used to deal with constraint violations. A descent sequence is generated which converges to a critical point under the Mangasarian–Fromovitz constraint qualification along with some other mild assumptions. The method is compared with a selection of existing methods on a suitable set of test problems. Keywords Multi-objective optimization · SQP method · Critical point · Mangasarian–Fromovitz constraint qualification · Purity metric · Spread metrics Mathematics Subject Classification 90C26 · 49M05 · 97N40 · 90B99
1 Introduction A widely used line search technique for solving constrained single objective optimization problems is SQP method, which was developed by Wilson in 1963 and modified by several researchers (see [16,36]) in various directions. A serious limitation of these methods is the inconsistency of the quadratic sub-problem. Powell ( [35]) suggested a modified sub-problem to overcome this restriction, which was further modified in [4,24,32] for better efficiency. SQP method in [4] converges to an infeasible point in some situations. SQP method in [25] is a two step method. But, SQP method in [32]
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Md Abu Talhamainuddin Ansary [email protected] Geetanjali Panda [email protected]
1
Department of Mathematics, Indian Institute of Technology Kharagpur, Khragpur, India
123
M. A. T. Ansary, G. Panda
is one step method and always converges to a feasible point. These developments are related to single objective optimization problems. In this article, a convergent SQP iterative scheme is developed for constrained multi-objective optimization problems, in the light of [32]. Classical methods for solving multi-objective optimization problems are either scalarization methods or heuristic methods. Scalarization methods reduce the multiobjective optimization problem to a single objective optimization problem using pre determined parameters. Heuristic methods do not guarantee the convergence to the solution. To address these limitations, line search methods for unconstrained multiobjective optimization problems have been developed since 2000 by many researchers ( [1,2,11,12,34]), which are treated as the extension of single objective line search techniques. Possible extension of these concepts to constrained multi-objective problems is an interesting area of research in recent times. The steepest descent method for multi-objec
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