In this paper, we show that higher-order optimality conditions can be obtain for arbitrary nonsmooth function. We introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper extended real function. This derivative is consistent with the classical higher-order Fréchet directional derivative in the sense that both derivatives of the same...

This paper focuses on master slave synchronization problem of piecewise affine (PWA) systems of continuous type. The synchronization problem is solved as a global asymptotic stability problem of the error dynamics between the master and the slave systems via a state feedback control law. Using a Lyapunov approach and the so-called S-procedure, new sufficient conditions are proposed. The synchro...

For a bilevel program with extremal value function, a necessary and sufficient condition for global optimality is given, which reduces the bilevel program to a max-min problem with linked constraints. Also, for the case where the extremal value function is polyhedral, this optimality condition gives the possibility of a resolution via a maximization problem of a polyhedral convex function over ...

In this paper we consider the strategy of stochastic bundling, which we define as a selling strategy in which firms guarantee a consumer will receive a good, but do not guarantee which one it will be. First we derive a new condition that ensures the optimality of stochastic bundling when there is no secondary market. Then we graphically consider the conditions under which stochastic bundling ma...

We establish alternative theorems for quadratic inequality systems. Consequently, we obtain Lagrange multiplier characterizations of global optimality for classes of non-convex quadratic optimization problems. We present a generalization of Dine’s theorem to a system of two homogeneous quadratic functions with a regular cone. The class of regular cones are cones K for which (K∪−K) is a subspace...

A convex semidefinite programming problem is a convex constrained optimization problem, where the constraints are linear matrix inequalities, for which the standard Lagrangian condition is sufficient for optimality. However, this condition requires constraint qualifications to completely characterize optimality. We present a necessary and sufficient condition for optimality without a constraint...

3 1 The term optimization 4 1.1 Global Optimization . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Why global optimization? . . . . . . . . . . . . . . . . . . . . 10 1.3 The structure of Global Optimization Algorithms . . . . . . . 12 1.4 Local Optimization of NLPs . . . . . . . . . . . . . . . . . . . 12 1.4.1 Notions of convex analysis . . . . . . . . . . . . . . . . 12 1.4.2 Necessary ...

In recent years, there has been an increasing interest in studying the develpoment of optimality conditions for nondifferentiable multiobjective programming problems. Many authors established and employed some different Kuhn and Tucker type necessary conditions or other type necessary conditions to research optimal solutions; see [1–27] and references therein. In [7], Lai and Ho used the Pareto...

We consider amathematical model in the form of a system of ordinary differential equations (ODE) foroptimally administrating cancer treatments. The ODE systemdynamics characterized by locating equilibrium points and stability propertiesare determined by linearization and usingappropriate Lyapunovfunctions. By applying optimal control theory, we seek to minimize the cost function associated with...

We develop sufficient optimality conditions for a Moreau-Yosida regularized optimal control problem governed by a semilinear elliptic PDE with pointwise constraints on the state and the control. We make use of the equivalence of a setting of Moreau-Yosida regularization to a special setting of the virtual control concept, for which standard second order sufficient conditions have been shown. Mo...