This paper deals with the study of a general class of nonlinear variational inequalities. An existence result is given, and a perturbed iterative scheme is analyzed for solving such problems.

We provide necessary and sufficient conditions for a minimal upper semicontinuous multifunction defined on a separable Banach space to be the subdifferential mapping of a Lipschitz function.

Consider a general class of constrained optimal control problems in canonical form. Using the classical control parameterization technique, the time (planning) horizon is partitioned into several subintervals. The control functions are approximated by piecewise constant or piecewise linear functions with pre-fixed switching times. However, if the optimal control functions to be obtained are piecewise continuous, the accuracy of this approximation process greatly depends on how fine the partition is. On the other hand, the performance of any optimization algorithm used is limited by the number of decision variables of the problem. Thus, the time horizon cannot be partitioned into arbitrarily many subintervals to reach the desired accuracy. To overcome this difficulty, the switching points should also be taken as decision variables. This is the main motivation of the paper. A novel transform, to be referred to as the control parameterization enhancing transform, is introduced to convert approximate optimal control problems with variable switching times into equivalent standard optimal control problems involving piecewise constant or piecewise linear control functions with pre-fixed switching times. The transformed problems are essentially optimal parameter selection problems and hence are solvable by various existing algorithms. For illustration, two non-trivial numerical examples are solved using the proposed method.

The paper revisits characterizations of strong stability and strong regularity of KarushKuhn-Tucker solutions of nonlinear programs with twice differentiable data. We give a unified framework to handle both concepts simultaneously.

In 1990 Gowda and Teboulle published the paper [16], making a comparison of several conditions ensuring the Fenchel-Rockafellar duality formula

inf{f(x) + g(Ax) | x ∈ X} = max{−f*(A*y*) − g*(− y*) | y* ∈ Y*}.

Probably the first comparison of different constraint qualification conditions was made by Hiriart-Urruty [17] in connection with ε-subdifferential calculus. Among them appears, as the basic sufficient condition, the formula for the conjugate of the corresponding function; such functions are: f1 + f2, g o A, max{fl,…, fn}, etc. In fact strong duality formulae (like the one above) and good formulae for conjugates are equivalent and they can be used to obtain formulae for ε-subdifferentials, using a technique developed in [17] and extensively used in [46].

We investigate the strict lower subdifferentiability of a real-valued function on a closed convex subset of Rn. Relations between the strict lower subdifferential, lower subdifferential, and the usual convex subdifferential are established. Furthermore, we present necessary and sufficient optimality conditions for a class of quasiconvex minimization problems in terms of lower and strict lower subdifferentials. Finally, a descent direction method is proposed and global convergence results of the consequent algorithm are obtained.

In management science and system engineering, problems with two incommensurate objectives are often detected. Bicriterion optimization finds an optimal solution for the problems. In this paper it is shown that bicriterion discrete optimal control problems can be solved by using a parametric optimization technique with relaxed convexity assumptions. Some necessary optimality conditions for discrete optimal control problems subject to a linear state difference equation are derived. It is shown that in this case no adjoint equation is required.

In this paper, we present generalizations of the Jacobian matrix and the Hessian matrix to continuous maps and continuously differentiable functions respectively. We then establish second-order optimality conditions for mathematical programming problems with continuously differentiable functions. The results also sharpen the corresponding results for problems involving C1.1-functions.