Nonsmooth Programming Research Articles

Mixed Duality without Constraint Qualification for Nonsmooth Fractional Programming

A class of nonsmooth multiobjective fractional programming is formulated. We establish the necessary and sufficient optimality conditions without the need of a constraint qualification. Then a mixed dual is introduced for a class of nonsmooth fractional programming problems, and various duality theorems are established without a constraint qualification.

$B$-SEMIPREINVEX FUNCTIONS AND VECTOR OPTIMIZATION PROBLEMS IN BANACH SPACES

In this paper, we extend the scalar-valued $B$-semipreinvex functions and vector-valued preinvex functions to the cases of vector-valued $B$-semipreinvex functions in Banach spaces. We investigate some properties for the vector-valued $B$-semipreinvex functions and consider a new class of vector-valued nonsmooth programming problems in which functions are locally Lipschitz. In terms of the Ralph vector sub-gradient, we obtain the generalized Kuhn-Tucker type sufficient optimality conditions and saddle point condition. Also, a generalized Mond-Weir type dual is formulated and some duality theorems are established involving locally Lipschitz $B$-semipreinvex functions for the pair of primal and dual programming. The results presented in this paper generalize some main results of Kuang and Batista Dos Santoset, Osuna-Gomez, Rojas-Medar and Rufian-Lizana.

B-SEMIPREINVEX FUNCTIONS AND VECTOR OPTIMIZATION PROBLEMS IN BANACH SPACES

In this paper, we extend the scalar-valued $B$-semipreinvex functions and vector-valued preinvex functions to the cases of vector-valued $B$-semipreinvex functions in Banach spaces. We investigate some properties for the vector-valued $B$-semipreinvex functions and consider a new class of vector-valued nonsmooth programming problems in which functions are locally Lipschitz. In terms of the Ralph vector sub-gradient, we obtain the generalized Kuhn-Tucker type sufficient optimality conditions and saddle point condition. Also, a generalized Mond-Weir type dual is formulated and some duality theorems are established involving locally Lipschitz $B$-semipreinvex functions for the pair of primal and dual programming. The results presented in this paper generalize some main results of Kuang and Batista Dos Santoset, Osuna-Gomez, Rojas-Medar and Rufian-Lizana.

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.

The approximation algorithm for solving a sort of non-smooth programming

An approximation algorithm for solving a sort of non-smooth programming is proposed. In the programming, the objective function is the Hölder function and the feasible region is contained in a rectangle (viz. hyper-rectangle). To establish the algorithm, the properties of the Hölder function and an approximation of the function by using Bernstein α-polynomial are studied. According to the properties of the approximation polynomial and the algorithm for solving geometric programming, the strategy for branching and bounding and the branch-and-bound algorithm are constructed to solve the programming. The convergence of the algorithm can be guaranteed by the exhaustive of the bisection of the rectangle. The feasibility of the algorithm is validated by solving an example.

A Sequential Smooth Penalization Approach to Mathematical Programs with Complementarity Constraints

In this paper, a mathematical program with complementarity constraints (MPCC) is reformulated as a nonsmooth constrained mathematical program via the Fischer–Burmeister function. Smooth penalty functions are used to treat this nonsmooth constrained program. Under linear independence constraint qualification, and upper level strict complementarity condition, together with some other mild conditions, we prove that the limit point of stationary points satisfying second-order necessary conditions of unconstrained penalized problems is a strongly stationary point, hence a B-stationary point of the original MPCC. Furthermore, this limit point also satisfies a second-order necessary condition of the original MPCC. Numerical results are presented to test the performance of this method.

Mean-variance-skewness model for portfolio selection with transaction costs

A mean-variance-skewness model is proposed for portfolio selection with transaction costs. It is assumed that the transaction cost is a V-shaped function of the difference between the existing portfolio and a new one. The mean-variance-skewness model is a non-smooth programming problem. To overcome the difficulty arising from non-smoothness, the problem was transformed into a linear programming problem. Therefore, this technique can be used to solve large-scale portfolio selection problems. A numerical example is used to illustrate that the method can be efficiently used in practice.

Wolfe type duality involving nonsmooth (B, η) -invex functions for a minmax programming problem

A nonsmooth minmax programming problem, under the assumption of proper (b, η) – invexity conditions on the functions involved in the objective and in the constraints, is considered. For such a problem, a sufficient optimality theorem is proved, a Wolfe type dual is presented and duality theorems relating the primal and the dual are proved. Applications of these results to certain nonsmooth fractional and nonsmooth generalized fractional programming duality are also presented.

LIPSCHITZ r-INVEX FUNCTIONS AND NONSMOOTH PROGRAMMING

ABSTRACT In the paper, a family of real Lipschitz functions, called r-invex, which represents a generalization of the notion of invexity is introduced. The principal analytic tool is the generalized gradient of Clarke for Lipschitz functions. Furthermore, under appropriate r-invexity assumptions, necessary optimality conditions of the Slater type and sufficient optimality conditions are obtained for a nonsmooth programming problem. Also some duality results are obtained for such problems.

On nonsmooth minimax programming problems containing V-ρ-invexity

We consider a class of nonsmooth minimax programming problems in which functions are locally Lipschitz. Necessary and sufficient optimality conditions and duality theorems for nonsmooth minimax programming problems are obtained under Vρ-invexity assumptions on objective and constraint functions

A Spectral Bundle Method for Semidefinite Programming

A central drawback of primal-dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically, semidefinite relaxations arising in combinatorial applications have sparse and well-structured cost and coefficient matrices of huge order. We present a method that allows us to compute acceptable approximations to the optimal solution of large problems within reasonable time. Semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems. These are convex nonsmooth programming problems and can be solved by bundle methods. We propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored to eigenvalue problems. Convergence follows from the traditional approach but a proof is included for completeness. We present numerical examples demonstrating the efficiency of the approach on combinatorial examples.

Finding Quadratic Schedules for Affine Recurrence Equations Via Nonsmooth Optimization

Frequently, affine recurrence equations can be scheduled more efficiently by quadratic scheduling functions than by linear scheduling functions. In this paper, the problem of finding optimal quadratic schedules for affine recurrence equations is formulated as a convex nonsmooth programming problem. In particular, sufficient constraints for causality are used generalizing Lamport's condition. In this way, the presented problem formulation becomes independent of the problem size. The research tool AQUAD is described implementing this problem formulation. Several nontrivial examples demonstrate that AQUAD can be effectively used to calculate quadratic schedules for affine recurrence equations. Finally, it is shown how array processors can be synthesized from affine recurrence equations which are scheduled by quadratic functions with a singular Hessian matrix.

The check of optimality conditions in nonsmooth programming

In this paper, the geometrical meaning of two basic optimality conditions for a class of common cases in nonsmooth programming is exposed. Thus it makes the check of optimality conditions for this class of nonsmooth programming problem much easier.

Optimality conditions and duality models for a class of nonsmooth continuous-time fractional programming problems

Both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonsmooth continuous-time programming problems. Subsequently, the structure and contents of these optimality results are utilized as a basis for constructing two parametric and four parameter-free duality models, and for proving weak, strong, and strict converse duality theorems. Furthermore, it is briefly pointed out how similar optimality and duality results can be derived for two special cases of the main problem containing arbitrary norms and square roots of positive semidefinite quadratic forms. These results generalize a number of similar results in the area of continuous-time programming, and subsume a great variety of cognate results originally developed for finite-dimensional nonlinear and fractional programming problems.

Lipschitz B-Vex Functions and Nonsmooth Programming

In this paper, the equivalence between the class of B-vex functions and that of quasiconvex functions is proved. Necessary and sufficient conditions, under which a locally Lipschitz function is B-vex, are established in terms of the Clarke subdifferential. Regularity of locally Lipschitz B-vex functions is discussed. Furthermore, under appropriate conditions, a necessary optimality condition of the Slater type and a sufficient optimality condition are obtained for a nonsmooth programming problem involving B-vex functions.

On saddle points and optima for non-smooth and non-convex programs

The equivalence between saddle points and optima and duality theorem (without constraint qualification) are established for a large class of non-smooth and non-convex programs

On generalised convex multi-objective nonsmooth programming

AbstractWe extend the concept of V-pseudo-invexity and V-quasi-invexity of multi-objective programming to the case of nonsmooth multi-objective programming problems. The generalised subgradient Kuhn-Tucker conditions are shown to be sufficient for a weak minimum of a multi-objective programming problem under certain assumptions. Duality results are also obtained.

On sufficiency and duality for generalised quasicnvex nonsmooth programs

For a nonsmooth multiobjective nonlinear program which involves inequality and equality constraints, Fritz John and Kuhn-Tucker type sufficient optimality conditions are obtained. Some duality theorems are also established for Mond-Weir type of duals. All results are given under generalized convexity assumptions.

Small convex-valued subdifferentials in mathematical programming

Various concepts of subdifferential for locally Lipschitz functions in finite dimensions are compared and a new characterization of a small convex-valued subdifferential is given A Fritz-John optimallty condition is derived for abstract nonsmooth programming problems in infinite dimensions uslng the Michel-Penot subdifferential and the concept of a strongly Lipschitz mapping

Generalized convex composite multi-objective nonsmooth programming and conditional proper efficiency

The concept of conditional proper efficiency has been incorporated to develop the duality theory for nonsmooth constrained multiobjective optimization problems where the objective functions. and the constraints are compositions of V-invex functions, V-pseudo-invex functions and V-quasi-invex functions and locally Lipschitz and Gateaux differentiable functions. Lagrangian necessary conditions. and new sufficient optirnality conditions for efficient and conditionally properly efficient solutions are presented. Some applications are also given