947 research outputs found
An interior-point method for mpecs based on strictly feasible relaxations.
An interior-point method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primaldual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty strictly feasible region. In contrast to previous approaches, the proposed relaxation scheme preserves the nonempty strict feasibility of each subproblem even in the limit. Local and superlinear convergence of the algorithm is proved even with a less restrictive strict complementarity condition than the standard one. Moreover, mechanisms for inducing global convergence in practice are proposed. Numerical results on the MacMPEC test problem set demonstrate the fast-local convergence properties of the algorithm
Optimality conditions and constraint qualifications for cardinality constrained optimization problems
The cardinality constrained optimization problem (CCOP) is an optimization
problem where the maximum number of nonzero components of any feasible point is
bounded. In this paper, we consider CCOP as a mathematical program with
disjunctive subspaces constraints (MPDSC). Since a subspace is a special case
of a convex polyhedral set, MPDSC is a special case of the mathematical program
with disjunctive constraints (MPDC). Using the special structure of subspaces,
we are able to obtain more precise formulas for the tangent and (directional)
normal cones for the disjunctive set of subspaces. We then obtain first and
second order optimality conditions by using the corresponding results from
MPDC. Thanks to the special structure of the subspace, we are able to obtain
some results for MPDSC that do not hold in general for MPDC. In particular we
show that the relaxed constant positive linear dependence (RCPLD) is a
sufficient condition for the metric subregularity/error bound property for
MPDSC which is not true for MPDC in general. Finally we show that under all
constraint qualifications presented in this paper, certain exact penalization
holds for CCOP
On the Burer-Monteiro method for general semidefinite programs
Consider a semidefinite program (SDP) involving an positive
semidefinite matrix . The Burer-Monteiro method uses the substitution to obtain a nonconvex optimization problem in terms of an
matrix . Boumal et al. showed that this nonconvex method provably solves
equality-constrained SDPs with a generic cost matrix when , where is the number of constraints. In this note we extend
their result to arbitrary SDPs, possibly involving inequalities or multiple
semidefinite constraints. We derive similar guarantees for a fixed cost matrix
and generic constraints. We illustrate applications to matrix sensing and
integer quadratic minimization.Comment: 10 page
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