Morgan A Hanson

On sufficiency of the Kuhn-Tucker conditions

If all the functions fo(x),f,(x),...,f,(x) are convex on C then these conditions are also sufficient. Various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming. Mangasarian [ 11 has speculated that pseudo-convexity of&(x) and quasiconvexity off(x) are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality. It will be shown that there are other wide classes of functions for which the conditions are sufficient. 545

Continuous time programming with nonlinear constraints

Abstract : A class of continous time nonlinear programming problems relating to problems of acquisition, stockpiling and distribution of materials is considered. Nonlinearity appears in both the objective function and the constraints. Necessary and sufficient conditions (Kuhn-Tucker conditions) are established and optimal solutions are characterized in terms of a duality theorem. (Author)

Continuous time programming with nonlinear time-delayed constraints

Abstract : A class of continuous time nonlinear programming problems relating to problems of acquisition, stockpiling, and distribution of materials is given. Nonlinearity appears in both the objective function and the constraints, and provision is made for time lags in the constraints. Necessary and sufficient conditions for the existence of solutions are established and optimal solutions are characterized in term of a duality theorem. (Author)

On nondifferentiable fractional minimax programming

Abstract Necessary and sufficient optimality conditions are derived for a nondifferentiable fractional minimax programming problem: Minimize x∈R n sup y∈Y {(f(x,y)+(x T Bx) 1/2 )/(h(x,y)−(x T Dx) 1/2 )} subject to g(x)⩽0, where Y is a compact subset of Rn; f(·,·):Rn×Rm→R and h(·,·):Rn×Rm→R are C1 on Rn×Rm; g(·):Rn→Rr is C1 on Rn; B and D are n×n symmetric positive semidefinite matrices. For this class of problems, two duals are proposed and weak, strong and strict converse duality theorems are established for each dual problem.

A sufficient condition for invexity

Abstract In nonlinear programming, invexity is sufficient for optimality (in conjunction with the Kuhn-Tucker conditions). In this paper we give a sufficient condition for invexity in nonlinear programming through the use of linear programming.

A simple solution of the van der Waerden permanent problem

The van der Waerden permanent problem was solved using mainly algebraic methods. A much simpler analytic proof is given using a new concept in optimization theory which may be of importance in the general theory of mathematical programming.

On the form of solutions to the linear continuous time programming problem and a conjecture by Tyndall

Abstract This paper gives the general form of solutions to the linear continuous time programming problem and shows that the solutions are piecewise smooth.

Solutions for a class of invex programming problems

A nonconvex constrained optimization problem is considered in which the constraints are of the form of generalized polynomials. An invexity kernel is established for this class of problem, and a consequent theorem gives sufficient conditions for the solutions of such problems.