Abstract
In this paper, we consider a sufficiently broad class of non-linear mathematical programs with disjunctive constraints, which, e.g. include mathematical programs with complemetarity/vanishing constraints. We present an extension of the concept of -stationarity which can be easily combined with the well-known notion of M-stationarity to obtain the stronger property of so-called -stationarity. We show how the property of -stationarity (and thus also of M-stationarity) can be efficiently verified for the considered problem class by computing -stationary solutions of a certain quadratic program. We consider further the situation that the point which is to be tested for -stationarity, is not known exactly, but is approximated by some convergent sequence, as it is usually the case when applying some numerical method.
Highlights
In this paper, we consider the following mathematical program with disjunctive constraints (MPDC) min f (x) x∈RnKi subject to Fi(x) ∈ Di := Dij, i = 1, . . . , mD, (1)j=1 where the mappings f : Rn → R and Fi : Rn → Rli, i = 1, . . . , mD are assumed to be continuously differentiable and Dij Denoting m :=⊂ Rli, mD i=1 li, j =, Ki, i mD are convex polyhedral sets.F := (F1, . . . , FmD ) : Rn → Rm, D := Di (2)i=1 we can rewrite the MPDC (1) in the form min f (x) subject to F(x) ∈ D
It is not known in general how to efficiently verify the M-stationarity conditions, since the description of the limiting normal cone involves some combinatorial structure which is not known to be resolved without enumeration techniques
For the disjunctive formulations of the problems mathematical program with complementarity constraints (MPCC) and mathematical program with vanishing constraints (MPVC) the Q- and QM-stationarity conditions have been worked out in detail in [14]. We extend this approach to the general problem MPDC
Summary
We consider the following mathematical program with disjunctive constraints (MPDC). M-stationarity requires only some weak constraint qualification but it does not preclude the existence of feasible descent directions It is not known in general how to efficiently verify the M-stationarity conditions, since the description of the limiting normal cone involves some combinatorial structure which is not known to be resolved without enumeration techniques. This algorithm either proves the existence of some feasible descent direction, i.e. the point is not B-stationary, or it computes multipliers fulfilling the QM-stationarity condition To this end, we consider quadratic programs with disjunctive constraints (QPDC), i.e. the objective function f in MPDC is a convex quadratic function and the mappings Fi, i = 1, . We denote by 0+C the recession cone of a convex set C
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