TY - GEN A1 - Mehlitz, Patrick T1 - Stationarity conditions and constraint qualifications for mathematical programs with switching constraints T2 - Mathematical Programming N2 - In optimal control, switching structures demanding at most one control to be active at any time instance appear frequently. Discretizing such problems, a so-called mathematical program with switching constraints is obtained. Although these problems are related to other types of disjunctive programs like optimization problems with complementarity or vanishing constraints, their inherent structure makes a separate consideration necessary. Since standard constraint qualifications are likely to fail at the feasible points of switching-constrained optimization problems, stationarity notions which are weaker than the associated Karush–Kuhn–Tucker conditions need to be investigated in order to find applicable necessary optimality conditions. Furthermore, appropriately tailored constraint qualifications need to be formulated. In this paper, we introduce suitable notions of weak, Mordukhovich-, and strong stationarity for mathematical programs with switching constraints and present some associated constraint qualifications. Our findings are exploited to state necessary optimality conditions for (discretized) optimal control problems with switching constraints. Furthermore, we apply our results to optimization problems with either-or-constraints. First, a novel reformulation of such problems using switching constraints is presented. Second, the derived surrogate problem is exploited to obtain necessary optimality conditions for the original program. KW - Constraint qualifications KW - Either-or-constraints KW - Nonlinear programming KW - Optimality conditions KW - Switching constraints Y1 - 2020 U6 - https://doi.org/10.1007/s10107-019-01380-5 SN - 0025-5610 SN - 1436-4646 VL - 181 IS - 1 SP - 149 EP - 186 ER - TY - GEN A1 - Kanzow, Christian A1 - Mehlitz, Patrick A1 - Steck, Daniel T1 - Relaxation schemes for mathematical programmes with switching constraints T2 - Optimization Methods and Software N2 - 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. KW - Constraint qualifications KW - mathematical programme with switching constraints KW - relaxation methods KW - global convergence Y1 - 2021 U6 - https://doi.org/10.1080/10556788.2019.1663425 SN - 1055-6788 SN - 1029-4937 VL - 36 IS - 6 SP - 1223 EP - 1258 ER - TY - GEN A1 - Mehlitz, Patrick T1 - On the linear independence constraint qualification in disjunctive programming T2 - Optimization N2 - Mathematical programmes with disjunctive constraints (MPDCs for short) cover several different problem classes from nonlinear optimization including complementarity-, vanishing-, cardinality- and switching-constrained optimization problems. In this paper, we introduce an abstract but reasonable version of the prominent linear independence constraint qualification which applies to MPDCs. Afterwards, we derive first- and second-order optimality conditions for MPDCs under validity of this constraint qualification based on so-called strongly stationary points. Finally, we apply our findings to some popular classes of disjunctive programmes and compare the obtained results to those ones available in the literature. Particularly, new second-order optimality conditions for mathematical programmes with switching constraints are by-products of our approach. KW - Constraint qualifications KW - disjunctive programming KW - linear independence constraint qualification KW - strong stationarity KW - second-order optimality conditions Y1 - 2020 U6 - https://doi.org/10.1080/02331934.2019.1679811 SN - 0233-1934 SN - 1029-4945 VL - Vol. 69 (2020) IS - 10 SP - 2241 EP - 2277 ER - TY - GEN A1 - Mehlitz, Patrick T1 - Asymptotic stationarity and regularity for nonsmooth optimization problems T2 - Journal of Nonsmooth Analysis and Optimization N2 - Based on the tools of limiting variational analysis, we derive a sequential necessary optimality condition for nonsmooth mathematical programs which holds without any additional assumptions. In order to ensure that stationary points in this new sense are already Mordukhovich-stationary, the presence of a constraint qualification which we call AM-regularity is necessary. We investigate the relationship between AM-regularity and other constraint qualifications from nonsmooth optimization like metric (sub-)regularity of the underlying feasibility mapping. Our findings are applied to optimization problems with geometric and, particularly, disjunctive constraints. This way, it is shown that AM-regularity recovers recently introduced cone-continuity-type constraint qualifications, sometimes referred to as AKKT-regularity, from standard nonlinear and complementarity-constrained optimization. Finally, we discuss some consequences of AM-regularity for the limiting variational calculus. KW - Asymptotic regularity KW - Asymptotic stationarity KW - Constraint qualifications KW - M-stationarity KW - Nonsmooth optimization KW - Variational analysis Y1 - 2020 U6 - https://doi.org/10.46298/jnsao-2020-6575 SN - 2700-7448 VL - 1 ER -