A loose Benders decomposition algorithm for approximating two-stage mixed-integer recourse models

We propose a new class of convex approximations for two-stage mixed-integer recourse models, the so-called generalized alpha-approximations. The advantage of these convex approximations over existing ones is that they are more suitable for efficient computations. Indeed, we construct a loose Benders decomposition algorithm that solves large problem instances in reasonable time. To guarantee the performance of the resulting solution, we derive corresponding error bounds that depend on the total variations of the probability density functions of the random variables in the model. The error bounds converge to zero if these total variations converge to zero. We empirically assess our solution method on several test instances, including the SIZES and SSLP instances from SIPLIB. We show that our method finds near-optimal solutions if the variability of the random parameters in the model is large. Moreover, our method outperforms existing methods in terms of computation time, especially for large problem instances.

The conditions of minimum for a smooth functionon the boundary of a quasidifferntiable set

In this paper, we consider problems of mathematical programming with nonsmoothconstraints of equality type given by quasidifferentiable functions. By using thetechnique of upper convex approximations, developed by B. N. Pshenichy, necessary conditionsof extremum for such problems are established. The Lagrange multipliers signsare specified by virtue of the fact that one can construct whole familers of upper convexapproximations for quasidifferentiable function and thus the minimum points in such extremalproblems are characterized more precisely. Also the simplest problem of calculusof variations with free right hand side is considered, where the left end of the trajectorystarts on the boundary of the convex set. The transversality condition at the left end of thetrajectory is improved provided sertain sufficient conditons hold