Optimality criteria form the foundations of mathematical programming both theoretically and computationally. In general, these criteria can be classified as either necessary or sufficient. Of course, one would like to have the same criterion be both necessary and sufficient. However, this occurs only under somewhat ideal conditions which are rarely satisfied in practice. In the absence of conve...

Under transactions costs, and generally any sort of friction, economic agents, acting dynamically in a stochastic environment, may find it optimal, in some region of the state space, to take no action at all. Action is triggered when the state of the economic system reaches the boundary of the ‘region of no action’. In this note, first-order conditions for the choice of the region of no action,...

We establish new necessary and sufficient optimality conditions for optimization problems. In particular, we establish tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints, such as box constraints, binary constraints, and simplex constraints. Our main theoretical tool for establishing these optimality conditions is ab...

We propose a deenition of \regular synthesis," more general than those suggested by other authors such as Boltyanskii and Brunovsk y, and an even more general notion of \regular presynthesis." We give a complete proof of the corresponding suuciency theorem, a slightly weaker version of which had been stated in an earlier article, with only a rough outline of the proof. We illustrate the strengt...

We present semidefinite relaxations of nonconvex, box-constrained quadratic programming, which incorporate the firstand second-order necessary optimality conditions, and establish theoretical relationships between the new relaxations and a basic semidefinite relaxation due to Shor. We compare these relaxations in the context of branch-and-bound to determine a global optimal solution, where it i...