A new optimality condition for minimization with general constraints is introduced. Unlike the KKT conditions, this condition is satissed by local minimizers of nonlinear programming problems, independently of constraint qualiications. The new condition implies, and is strictly stronger than, Fritz-John optimality conditions. Suu-ciency for convex programming is proved.

The support vector machine (SVM) is a powerful and widely used classification algorithm. This paper uses the Karush-Kuhn-Tucker conditions to provide rigorous mathematical proof for new insights into behavior of SVM. These perhaps unexpected relationships between SVM two other linear classifiers: mean difference maximal data piling direction. For example, we show that in many cases can be viewe...

The lecture presents an integrated modeling and solution framework aiming at the robust and efficient solution of very large instances of tree-sparse programs. This wide class of nonlinear programs (NLP) is characterized by an underlying tree topology. It includes, in particular, dynamic stochastic programs in scenario tree formulation, multistage stochastic programs, where the objective and co...

To date the primary focus of most constrained approximate optimization strategies is that application of the method should lead to improved designs. Few researchers have focused on the development of constrained approximate optimization strategies that are assured of converging to a Karush-Kuhn-Tucker (KKT) point for the problem. Recent work by the authors based on a trust region model manageme...

This paper focuses on a time-varying constrained nonconvex optimization problem, and considers the synthesis analysis of online regularized primal-dual gradient methods to track Karush--Kuhn--Tucker (KKT) trajectory. The proposed method is implemented in running fashion, sense that underlying problem changes during execution algorithms. In order study its performance, we first derive continuous...

Multistage stochastic programs can be seen as discrete optimal control problems with a characteristic dynamic structure induced by the scenario tree. To exploit that structure, we propose a highly eecient dynamic programming recursion for the computationally intensive task of KKT systems solution within an interior point method. Test runs on a multistage portfolio selection problem demonstrate ...

In this paper, we investigate the application of feasible direction method for an optimistic nonlinear bilevel programming problem. The convex lower level problem of an optimistic nonlinear bilevel programming problem is replaced by relaxed KKT conditions. The feasible direction method developed by Topkis and Veinott [22] is applied to the auxiliary problem to get a Bouligand stationary point f...

In this paper we give necessary optimality conditions of Fritz-John and Kuhn-Tucker (KKT) conditions for non-linear infinite dimensional programming problem with operatorial constraints. We use an alternative theorem. Some of the known results in finite dimensional case have been extended to infinite dimensional case with suitable conditions.

In many seqential quadratic programming algorithms for constrained optimization the calculation of an effective search direction depends on the (estimated) Hessian of the Lagrangian being positive definite on the null space of the active constraints. This paper reports some numerical experience with two techniques for checking the properties of the Hessian and, if necessary, modifying it during...