A PVT-TYPE ALGORITHM FOR MINIMIZING A NONSMOOTH CONVEX FUNCTION

A general framework of the (parallel variable transformation) PVT-type algorithm, called the PVT-MYR algorithm, for minimizing a nonsmooth convex function is proposed, via the Moreau-Yosida regularization. As a particular scheme of this framework an ε-scheme is also presented. The global convergence of this algorithm is given under the assumptions of strong convexity of the objective function and an ε-descent condition determined by an ε-forced function. An appendix stating the proximal point algorithm is recalled in the last section

Exponential Stabilizability of Switched Systems with Polytopic Uncertainties

The exponential stabilizability of switched nonlinear systems with polytopic uncertainties is explored by employing the methods of nonsmooth analysis and the minimum quadratic Lyapunov function. The switchings among subsystems are dependent on the directional derivative along the vertex directions of subsystems. In particular, a sufficient condition for exponential stabilizability of the switched nonlinear systems is established considering the sliding modes and the directional derivatives along sliding modes. Furthermore, the matrix conditions of exponential stabilizability are derived for the case of switched linear system and the numerical example is given to show the validity of the synthesis results

A modified Brown algorithm for solving singular nonlinear systems with rank defects

AbstractA modified Brown algorithm for solving a class of singular nonlinear systems, F(x)=0, where x,F∈Rn, is presented. This method is constructed by combining the discreted Brown algorithm with the space transforming method. The second-order information of F(x) at a point is not required calculating, which is different from the tensor method and the Hoy's method. The Q-quadratic convergence of this algorithm and some numerical examples are given as well

A superlinear space decomposition algorithm for constrained nonsmooth convex program

AbstractA class of constrained nonsmooth convex optimization problems, that is, piecewise C2 convex objectives with smooth convex inequality constraints are transformed into unconstrained nonsmooth convex programs with the help of exact penalty function. The objective functions of these unconstrained programs are particular cases of functions with primal–dual gradient structure which has connection with VU space decomposition. Then a VU space decomposition method for solving this unconstrained program is presented. This method is proved to converge with local superlinear rate under certain assumptions. An illustrative example is given to show how this method works

A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints

To reflect uncertain data in practical problems, stochastic versions of the mathematical program with complementarity constraints (MPCC) have drawn much attention in the recent literature. Our concern is the detailed analysis of convergence properties of a regularization sample average approximation (SAA) method for solving a stochastic mathematical program with complementarity constraints (SMPCC). The analysis of this regularization method is carried out in three steps: First, the almost sure convergence of optimal solutions of the regularized SAA problem to that of the true problem is established by the notion of epiconvergence in variational analysis. Second, under MPCC-MFCQ, which is weaker than MPCC-LICQ, we show that any accumulation point of Karash-Kuhn-Tucker points of the regularized SAA problem is almost surely a kind of stationary point of SMPCC as the sample size tends to infinity. Finally, some numerical results are reported to show the efficiency of the method proposed

Flow shop scheduling with deteriorating jobs under dominating machines

This paper addresses no-wait or no-idle flow shop scheduling problems with deteriorating jobs, i.e., jobs whose processing times are an increasing function of their starting time. A simple linear deterioration function is assumed and some dominating relationships between machines can be satisfied. It is shown that for the problems to minimize makespan or weighted sum of completion time, polynomial algorithms still exist, although these problems are more complicated than the classical ones. When the objective is to minimize maximum lateness or maximum tardiness, the solutions of a classical version may not hold.Scheduling Flow shop Simple linear deterioration Dominating machines