Weak Pareto optimality of multiobjective problems in a locally convex linear topological space

In this study, the Dubovitskii-Milyutin type optimization theory is extended to multiobjective programs in a locally convex linear topological space, producing necessary conditions for a weak Pareto optimum. In the case of an ordinary multiobjective convex program, generalized Kuhn-Tucker conditions by a subdifferential formula are necessary and sufficient for a weak Pareto optimum.The author is grateful for the useful suggestions and comments of Professor N. Furukawa and the referee.     

Nonself-Adjoint Operators with Almost Hermitian Spectrum: Weak Annihilators

We consider nonself-adjoint nondissipative trace class additive perturbations L=A+iV of a bounded self-adjoint operator A in a Hilbert space ,H. The main goal is to study the properties of the singular spectral subspace N _i ⁰ of L corresponding to part of the real singular spectrum and playing a special role in spectral theory of nonself-adjoint nondissipative operators.To some extent, the properties of N _i ⁰ resemble those of the singular spectral subspace of a self-adjoint operator. Namely, we prove that L and the adjoint operator ,L ^* are weakly annihilated by some scalar bounded outer analytic functions if and only if both of them satisfy the condition N _i ⁰ =H. This is a generalization of the well-known Cayley identity to nonself-adjoint operators of the above-mentioned class.     

Optimality and Duality in Nondifferentiable and Multiobjective Programming under Generalized d-Invexity

In this paper, we are concerned with the nondifferentiable multiobjective programming problem with inequality constraints. We introduce four new classes of generalized d-type-I functions. By utilizing the new concepts, Antczak type Karush-Kuhn-Tucker sufficient optimality conditions, Mond-Weir type and general Mond-Weir type duality results are obtained for non-differentiable and multiobjective programming.     

Second-order sufficient optimality conditions for local and global nonlinear programming

This paper presents a new approach to the sufficient conditions of nonlinear programming. Main result is a sufficient condition for the global optimality of a Kuhn-Tucker point. This condition can be verified constructively, using a novel convexity test based on interval analysis, and is guaranteed to prove global optimality of strong local minimizers for sufficiently narrow bounds. Hence it is expected to be a useful tool within branch and bound algorithms for global optimization.     

Supplementary optimality properties of the restoration phase of sequential gradient-restoration algorithms for optimal control problems

In this paper, sequential gradient-restoration algorithms for optimal control problems are considered, and attention is focused on the restoration phase. It is shown that the Lagrange multipliers associated with the restoration phase not only solve the auxiliary minimization problem of the restoration phase, but are also endowed with a supplementary optimality property: they minimize a special functional, quadratic in the multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.Dedicated to L. CesariThis work was supported by a grant of the National Science Foundation.     

On the existence of overtaking trajectories

A study is made of the conditions sufficient for the existence of overtaking trajectories for a class of infinite time-horizon, time-variant optimal control systems. Nonautonomy is restricted to disturbances with limits at infinity. The convergence property of the overtaking trajectories to the optimal steady-state limit is shown.     

Second-Order Global Optimality Conditions for Optimization Problems

Second-order optimality conditions are studied for the constrained optimization problem where the objective function and the constraints are compositions of convex functions and twice strictly differentiable functions. A second-order sufficient condition of a global minimizer is obtained by introducing a generalized representation condition. Second-order minimizer characterizations for a convex program and a linear fractional program are derived using the generalized representation condition     

Optimal control of systems governed by time-delayed,second-order,linear, parabolic partial differential equations with a first boundary condition

In this paper, we consider a class of systems governed by time-delayed, second-order, linear, parabolic partial differential equations with first boundary conditions. The existence and uniqueness of solutions of this class of systems are established in Theorem 3.2. A necessary condition for optimality for the corresponding controlled system is presented in Theorem 5.1. For the proof of this theorem, we develop several preparatory results in Sections 2, 3, and 4.