Journal
OPTIMIZATION METHODS & SOFTWARE
Volume 36, Issue 6, Pages 1223-1258Publisher
TAYLOR & FRANCIS LTD
DOI: 10.1080/10556788.2019.1663425
Keywords
Constraint qualifications; mathematical programme with switching constraints; relaxation methods; global convergence
Categories
Funding
- Deutsche Forschungsgemeinschaft - DFG [KA 1296/24-1, SPP 1962]
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This paper investigates the solution methods for switching-constrained optimization problems by adapting relaxation methods from numerical mathematical programs. Detailed convergence analysis is provided for these adapted relaxation schemes. It is shown that certain methods are capable of identifying stationary points of switching-constrained programs under suitable assumptions.
Switching-constrained optimization problems form a difficult class of mathematical programmes since their feasible set is almost disconnected while standard constraint qualifications are likely to fail at several feasible points. That is why the application of standard methods from nonlinear programming does not seem to be promising in order to solve such problems. In this paper, we adapt several relaxation methods which are well known from the numerical treatment of mathematical programmes with complementarity constraints to the setting of switching-constrained optimization. A detailed convergence analysis is provided for the adapted relaxation schemes of Scholtes as well as Kanzow and Schwartz. While Scholtes' method and the relaxation scheme of Steffensen and Ulbrich only find weakly stationary points in general, it is shown that the adapted relaxation scheme of Kanzow and Schwartz is capable of identifying Mordukhovich-stationary points of switching-constrained programmes under suitable assumptions. Some computational experiments and a numerical comparison of the proposed methods based on examples from logical programming, switching control, and portfolio optimization close the paper.
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