Polaroids: A new tool in non-convex and in integer programming

This paper presents a generalization, called polaroid, of the concept of polar sets A list of properties satisfied by polaroids is established indicating that the new concept nay be fruitfully used in an area of non-convex (called here polar) programming as well as in integer programming, by means of polaroid cuts; this class of new cuts contains the ones defined by Tuy for concave programming (a special case of polar programming) and by Balas integer programming; it furthermore provides for new degrees of freedom in the construction of algorithms in the above-mentioned areas of mathematical programming.     

Elements of a theory in non-convex programming

The question of necessary and sufficient optimality conditions for non-convex programs is analyzed in the general context of subadditivity. Several types of convex set extensions are investigated to generate valid inequalities from the corresponding gauge functions.     

Convex and polaroid extensions

In an effort towards a comprehensive and unified theory, this note presents some new results in the area of non-convex programming within the framework of convex (sets and function) analysis. The entire study is primarily devoted to the development of useful tools for extreme point programs (such as concave or integer programs).