Nonsmooth Nonlinear Programming Research Articles

A Nonsmooth Nonlinear Programming Based Predictive Control for Mechanical Servo Systems with Backlash‐like Hysteresis

AbstractIn this paper, a multi‐step‐ahead predictive control approach for dynamic systems with preceded backlash‐like hysteresis based on nonsmooth nonlinear programming is proposed. In this approach, a nonsmooth multi‐step‐ahead predictive model is developed for long‐range prediction of the controlled dynamic systems with preceded backlash‐like hysteresis. Then, the predictive control strategy is treated as a problem of on‐line nonsmooth nonlinear programming. Subsequently, the stability of the nonsmooth predictive control system is analyzed and the corresponding stability condition is derived. Afterward, a numerical example and a simulation based on a mechanical servo system are presented, respectively.

On the functions with pseudoconvex sublevel sets and optimality conditions

A new class of generalized convex functions, called the functions with pseudoconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex. Optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality, equality and a set constraints are obtained in terms of the lower Hadamard directional derivative. In particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets.

Global Convergence of a Robust Smoothing SQP Method for Semi-Infinite Programming

The semi-infinite programming (SIP) problem is a program with infinitely many constraints. It can be reformulated as a nonsmooth nonlinear programming problem with finite constraints by using an integral function. Due to the nondifferentiability of the integral function, gradient-based algorithms cannot be used to solve this nonsmooth nonlinear programming problem. To overcome this difficulty, we present a robust smoothing sequential quadratic programming (SQP) algorithm for solving the nonsmooth nonlinear programming problem. At each iteration of the algorthm, we need to solve only a quadratic program that is always feasible and solvable. The global convergence of the algorithm is established under mild conditions. Numerical results are given.

Generalized Neural Network for Nonsmooth Nonlinear Programming Problems

In 1988 Kennedy and Chua introduced the dynamical canonical nonlinear programming circuit (NPC) to solve in real time nonlinear programming problems where the objective function and the constraints are smooth (twice continuously differentiable) functions. In this paper, a generalized circuit is introduced (G-NPC), which is aimed at solving in real time a much wider class of nonsmooth nonlinear programming problems where the objective function and the constraints are assumed to satisfy only the weak condition of being regular functions. G-NPC, which derives from a natural extension of NPC, has a neural-like architecture and also features the presence of constraint neurons modeled by ideal diodes with infinite slope in the conducting region. By using the Clarke's generalized gradient of the involved functions, G-NPC is shown to obey a gradient system of differential inclusions, and its dynamical behavior and optimization capabilities, both for convex and nonconvex problems, are rigorously analyzed in the framework of nonsmooth analysis and the theory of differential inclusions. In the special important case of linear and quadratic programming problems, salient dynamical features of G-NPC, namely the presence of sliding modes , trajectory convergence in finite time, and the ability to compute the exact optimal solution of the problem being modeled, are uncovered and explained in the developed analytical framework.

Second-order necessary conditions for optimality in nonsmooth nonlinear programming

The notion of a second-order directional derivative introduced by Ben-Tal and Zowe and a generalization of this notion by Penot are used to state new second-order necessary optimality conditions for a nonsmooth nonlinear programming problem with a finite number of inequality and equality constraints

Second-Order Optimality Conditions in Nonlinear Programming Obtained by Way of Epi-Derivatives

Dedicated to the memory of Robin W. Chaney Second-order optimality conditions for finite-dimensional smooth and nonsmooth nonlinear programming are obtained by a new method that emphasizes a close connection with geometrical approximation of the essential objective function. The approximation is secured by the use of certain epi-derivatives defined by epiconvergence. The optimality conditions are expressed in a form that covers general interval constraints and their possible representation through penalties or an augmented Lagrangian. An abstract constraint involving restriction to a convex polyhedron is incorporated.

Second-Order Sufficient Conditions in Nonsmooth Optimization

Second-order conditions are given which are sufficient to guarantee that a given point be a local solution to certain types of finite-dimensional nonsmooth nonlinear programming problems. Both unconstrained and constrained problems are considered. The conditions of the sufficiency theorems presented here are expressed in terms of the generalized gradients of nonsmooth analysis and (with one exception) certain second-order directional derivatives. These sufficiency theorems are complementary to some necessity theorems, which are proved elsewhere.

Second-Order Necessary Conditions in Constrained Semismooth Optimization

First- and second-order conditions are given which must necessarily be satisfied by local minimizers for certain finite-dimensional nonsmooth nonlinear programming problems. The problems considered are of standard form, having a finite number of equality and inequality constraints. The principal result does not require a constraint qualification, but does require that the functions be semismooth at the minimizer. The necessary conditions are stated in terms of the generalized gradients of nonsmooth analysis and certain second-order directional derivatives.

A General Sufficiency Theorem for Nonsmooth Nonlinear Programming

A General Sufficiency Theorem for Nonsmooth Nonlinear Programming

A general sufficiency theorem for nonsmooth nonlinear programming

Second-order conditions are given which are sufficient to guarantee that a given point be a local minimizer for a real-valued locally Lipschitzian function over a closed set in n n -dimensional real Euclidean space. These conditions are expressed in terms of the generalized gradients of Clarke. The conditions provide a very general and unified framework into which many previous first- and second-order theorems fit.