Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Here, nonsmooth approximate gradient projection and complementary approximate Karush-Kuhn-Tucker conditions are presented. These sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constrai...

This paper presents a dual of a general linear fractional functions programming problem. Dual is shown to be a linear programming problem. Along with other duality theorems, complementary slackness theorem is also proved. A simple numerical example illustrates the result.

The tutorial is an introduction to programming today’s PC graphics hardware. It covers basic hardware architecture, optimization, programming interfaces and languages. It presents graphics and non-graphics applications. While the tutorial assumes basic knowledge in programming and principles of 3D computer graphic, familiarity with PC graphics hardware is unnecessary. The tutorial notes below a...

The role of pair programming process in software development is controversial. This controversy arises in part from their being presented as alternatives, yet it would be more helpful to see them as complementary software management tools. This paper describes the application of such a complementary model, software process fusion (SPF), in a real-world software management situation in China. Pa...

We define the primal and dual linear programming problems involving interval numbers as the way of traditional linear programming problems. We discuss the solution concepts of primal and dual linear programming problems involving interval numbers without converting them to classical linear programming problems. By introducing new arithmetic operations between interval numbers, we prove the weak...

Arising from research in the computer science community, constraint programming is a relatively new technique for solving optimization problems. For those familiar with mathematical programming, there are a number of language barriers that exist that make it difficult to understand the concepts of constraint programming. This paper, which assumes a minimal background in linear and integer progr...