Many engineering optimization problems involve a special kind of discrete variable that can be represented by a number, but this representation has no significance. Such variables arise when a decision involves some situation like a choice from an unordered list of categories. This has two implications: The standard approach of solving problems with continuous relaxations of discrete variables ...

Conjugate gradient methods are an important class of methods for unconstrained optimization, especially for large-scale problems. Recently, they have been much studied. This paper proposes a three-parameter family of hybrid conjugate gradient methods. Two important features of the family are that (i) it can avoid the propensity of small steps, namely, if a small step is generated away from the ...

For a locally optimal solution to the nonlinear semidefinite programming problem, under Robinson’s constraint qualification, the following conditions are proved to be equivalent: the strong second order sufficient condition and constraint nondegeneracy; the nonsingularity of Clarke’s Jacobian of the Karush-Kuhn-Tucker system; the strong regularity of the Karush-Kuhn-Tucker point; and others.

Abstrac t . There are well-known first-order sufficient conditions for a point x o to be a strict locally optimal solution of a nonlinear programming problem. In this paper, we show that these conditions also guarantee that Xo is an isolated stationary point of the considered program provided a constraint qualification holds. This result has an interesting application to finite convergence of a...

A bstract We study the ? -invariant of a Dirac operator on manifold with boundary subject to local conditions help heat kernel methods. In even dimensions, we relate this invariant -invariants operator, while in odd dimension, it is expressed through index operators. stress necessity strong ellipticity condition for applicability our show that Witten-Yonekura are not strongly elliptic, though t...