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In the present paper, we consider the inequality constrained vector problem with continuously Fréchet differentiable objective functions and constraints. We obtain second-order necessary optimality conditions of Karush-Kuhn-Tucker type for weak efficiency. A new second-order constraint qualification of Zangwill type is introduced. It is applied in the optimality conditions. Some connections with other constraint qualifications are established. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to draw the attention of the nonsmooth analysis and mathematical finance communities to the scope for applications of nonsmooth optimization to finance by studying in detail two illustrative examples. The first concerns the maximization of a terminal utility function in an investment problem with transaction costs. The second concerns the calculation of the duration of a bond for general term structures of interest rates. The emphasis is on methodology. Key Words. Nonsmooth optimization; utility maximization; transaction costs; bond duration; general term structure changes. [ABSTRACT FROM AUTHOR]

In this paper, we consider a constrained nonconvex nonsmooth optimization, in which both objective and constraint functions may not be convex or smooth. With the help of the penalty function, we transform the problem into an unconstrained one and design an algorithm in proximal bundle method in which local convexification of the penalty function is utilized to deal with it. We show that, if adding a special constraint qualification, the penalty function can be an exact one, and the sequence generated by our algorithm converges to the KKT points of the problem under a moderate assumption. Finally, some illustrative examples are given to show the good performance of our algorithm. [ABSTRACT FROM AUTHOR]

We consider a convexification method for a class of nonsmooth monotone functions. Specifically, we prove that a semismooth monotone function can be converted into a convex function via certain convexification transformations. The results derived in this paper lay a theoretical base to extend the reach of convexification methods in monotone optimization to nonsmooth situations. [ABSTRACT FROM AUTHOR]

We consider the proximal form of a bundle algorithm for minimizing a nonsmooth convex function, assuming that the function and subgradient values are evaluated approximately. We show how these approximations should be controlled in order to satisfy the desired optimality tolerance. For example, this is relevant in the context of Lagrangian relaxation, where obtaining exact information about the function and subgradient values involves solving exactly a certain optimization problem, which can be relatively costly (and as we show, in any case unnecessary). We show that approximation with some finite precision is sufficient in this setting and give an explicit characterization of this precision. Alternatively, our result can be viewed as a stability analysis of standard proximal bundle methods, as it answers the following question: for a given approximation error, what kind of approximate solution can be obtained and how does it depend on the magnitude of the perturbation? Key Words. Nonsmooth optimization; convex optimization; bundle methods; stability analysis; perturbations. [ABSTRACT FROM AUTHOR]

We present a survey on the results related to the theory of lexicographic differentiation. This theory ensures an efficient computation of generalized ( lexicographic) derivative of a nonsmooth function belonging to a special class of lexicographically smooth functions. This class is a linear space which contains all differentiable functions, all convex functions, and which is closed with respect to component-wise composition of the members. In order to define lexicographic derivative in a unique way, it is enough to fix a basis in the space of variables. Lexicographic derivatives can be used in black-box optimization methods. We give some examples of applications of these derivatives in analysis of nonsmooth functions. It is shown that the system of lexicographic derivatives along a fixed basis correctly represents corresponding nonsmooth function (Newton-Leibnitz formula). We present nonsmooth versions of standard theorems on potentiality of nonlinear operators, on differentiation of parametric integrals and on differentiation of functional sequences. Finally, we show that an appropriately defined lexicographic subdifferential ensures a more rigorous selection of a candidate optimal solution than the subdifferential of Clarke. [ABSTRACT FROM AUTHOR]

We present a general iterative procedure for solving generalized equations in the nonsmooth framework. To this end, we consider a class of functions admitting a certain type of approximation and establish a local convergence theorem that one can apply to a wide range of particular problems. [ABSTRACT FROM AUTHOR]

Studies an example corresponding to the nonsmooth case. Determination of an optimal control and optimal trajectories that give a minimum value to a certain functional; Indication of the one-parameter symmetry group of the problem; Existence of the solution; Investigation of the properties of the optimal synthesis; Solution of the Hamilton-Jacobi equation for the auxiliary problem.