Numerical Methods for Mathematical Programs with Disjunctive Constraints
Mag. Matúš BenkoFeb. 9, 2017, noon S3 048
Motivated by an increasing interest in mathematical programs with complementary constraints (MPCCs) and mathematical programs with vanishing constraints (MPVCs), in this thesis we consider a generalization of these programs, the so-called mathematical programs with disjunctive constraints (MPDCs).
In the first part of the thesis we develop new stationarity concepts called Q- and Q_M-stationarity and discuss their properties. First, we define Q- and Q_M-stationarity for general mathematical programs, i.e. without the assumption of disjunctive structure, and compare them to the well known B-stationarity. Then we concentrate on applications to MPCCs, MPVCs and MPDCs, where we estimate the regular normal cone to the feasible set and consequently we obtain conditions securing S-stationarity. In case of MPCCs and MPVCs, the obtained conditions are milder than the conditions that appear in literature, especially for MPVCs.
Next we extend the concepts of Q- and Q_M-stationarity and we apply them to a wider class of MPDCs. As a result, we obtain an algorithm, based on Q-stationarity, for verification of M- or Q_M-stationarity of a point for MPDCs. Moreover, we present some results for numerical methods for solving MPDCs which prevent convergence to non M-stationary and non Q_M-stationary points.
In the second part of the thesis we propose an SQP algorithm for MPDCs which solves at each iteration a quadratic program with linear disjunctive constraints, a so-called auxiliary problem. We show that all limit points of the sequence of iterates generated by the SQP method are at least M-stationary, provided we can find at least M-stationary solutions of the auxiliary problem. We mention that these convergence results can be improved to guarantee the stronger property of Q_M-stationarity of the limit points. Next we demonstrate how to solve the auxiliary problem in special cases of MPCCs and MPVCs. In case of MPVCs, the algorithm is based on the concept of Q-stationarity and can be generalized to the case of MPDCs. We conclude the thesis by numerical results.