Pseudo-duality and saddle points

Results associated with saddle-type stationary points are described. It is shown that barrier-type functions are pseudo-duals of generalized Lagrangian functions, while augmented Lagrangians are pseudo-duals of the regular Lagrangian function. An application of pseudo-duality to a min-max problem is illustrated, together with several other examples.     

Effects of the three-spin interactions on the magnetic properties of a simple cubic Ising ferromagnet

For a spin one-half simple cubic Ising ferromagnet, the three-spin interaction are shown to have significant effects on the Curie temperature, the magnetization and the magnetic specific heat based on a calculation by the Bethe-Peierls theory.     

A duality approach to minimax results for quasi-saddle functions in finite dimensions

In this paper we show how saddle point theorems for a quasiconvex—quasiconcave function can be derived from duality theory. A symmetric duality framework that provides the machinery for deriving saddle point theorems is presented. Generating the theorems,via the framework, provides a deeper understanding of assumptions employed in existing theorems which do not utilize duality theory.     

Secant relations versus positive definiteness in quasi-Newton methods

The effect of using a nonsymmetric and nonpositive-definite matrix for approximation of the Hessian inverse in unconstrained optimization is investigated. To this end, a new algorithm, which may be viewed as a member of the Huang family, is derived. The proposed algorithm possesses the quadratic termination property without exact line search. It seems from the numerical results that it is not essential to use a symmetric and positive-definite matrix.     

Conjugacy in quasi-convex programming

This paper develops a symmetric conjugate relation for quasi-convex functions. The concept of an evenly quasi-convex function is introduced and it is shown that this is the required property for a duality framework in quasi-convex programming.     

Global solutions of mathematical programs with intrinsically concave functions

An implicit enumeration technique for solving a certain type of nonconvex program is described. The method can be used for solving signomial programs with constraint functions defined by sums of quasiconcave functions and other types of programs with constraint functions called intrinsically concave functions. A signomial-type example is solved by this method. The algorithm is described together with a convergence proof. No computational results are available at present.     

Pseudo-duality in mathematical programs with quotients and products

A new approach to duality in fractional programs is described. The techniques used are based on the theory of pseudoduality. The results are generalized to include mathematical programs with quotients and products of finitely many functionals. The duals of the linear and quadratic cases are explicitly calculated, demonstrating the power of pseudo-duality theory.The author wishes to thank Prof. S. Schaible for his helpful comments. In discussions with Prof. Schaible, it was found that the duals for the linear and the quadratic cases are identical with those derived in Ref. 3.     

Local-global properties of bifunctions

Representations of composite systems, such as bilinear programming, models of consumer/producer behavior, and sensitivity problems involve bifunctions (functions of two vector arguments). Such bifunctions are typically convex, pseudoconvex, or quasiconvex in each of their arguments, but not jointly convex, pseudoconvex, or quasiconvex. These functions do not in general possess the strong local-global property, namely, that every stationary point is a global minimum. In this paper, we define conditions that ensure that a bifunction possesses only a global minimum. In exploring this question, we use P-convexity and pseudo P-convexity, which are classes of bifunctions that generalize quasiconvexity and pseudoconvexity.     

Numerical computation of second-order system response with variable parameters

Zusammenfassung Die im Aufsatz vorgeschlagene Methode erlaubt die numerische Berechnung des dynamischen Verhaltens von Systemen zweiter Ordnung mit veränderlichen Parametern. Dazu wird das System in zwei Teilsysteme erster Ordnung aufgespalten, die in Reihe wirken. Ihr Verhalten in aufeinanderfolgenden kurzen Zeitintervallen ist dann leicht zu berechnen. Als Beispiele für die Anwendung der Methode wurden Systeme mit veränderlicher hydraulischer Dämpfung gewählt. Eine Erweiterung auf andere Dämpfungsarten und auf Fälle veränderlicher Eigenfrequenz, wie auch auf Systeme höherer Ordnung, ist ohne weiteres möglich.