Last lecture we introduce linear programs (LP) and saw how to model problems with LPs. In this lecture we study duality of LPs. Roughly speaking, the dual of a minimization LP is a maximization LP which its solutions provide lower bounds for the optimum of the original LP. First we show how to construct dual of an LP. Then we talk about relation between the optimum of an LP and its dual. In par...

We convert the self-dual model of Townsend, Pilch, and Nieuwenhuizen to a first-class system using the generalized canonical formalism of Batalin, Fradkin, and Tyutin and show that gauge-invariant fields in the embedded model can be identified with observables in the Maxwell-Chern-Simons theory as well as with the fundamental fields of the self-dual model. We construct the phase-space partition...

We construct dual descriptions of (0, 2) gauged linear sigma models. In some cases, the dual is a (0, 2) Landau-Ginzburg theory, while in other cases, it is a non-linear sigma model. The duality map defines an analogue of mirror symmetry for (0, 2) theories. Using the dual description, we determine the instanton corrected chiral ring for some illustrative examples. This ring defines a (0, 2) ge...

It has recently been proposed that certain nonsupersymmetric type II orbifolds have vanishing perturbative contributions to the cosmological constant. We show that techniques of Sen and Vafa allow one to construct dual type II descriptions of these models (some of which have no weakly coupled heterotic dual). The dual type II models are given by the same orbifolds with the string coupling S and...

In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if the optimal control satisfies an exponential moment condition or if the value function is continuous on the closure of its domain. The key idea is to ...

We reprove that the approximate degree of the OR function on n bits is Ω( √ n). We consider a linear program which is feasible if and only if there is an approximate polynomial for a given function, and apply the duality theory. The duality theory says that the primal program has no solution if and only if its dual has a solution. Therefore one can prove the nonexistence of an approximate polyn...

The paper presents an important step toward the complete classification of all optimal binary self-dual codes of length 52 that possess an automorphism of odd prime order. Using a method for constructing and classifying binary self-dual codes with an automorphism of odd prime order p we give full classification of all [52, 26, 10] binary self-dual codes with an authomorphism of type 3-(16,4) an...

This paper deals with topological design of a hie~rchical two-levef network where the upper-level hub network is of ring type and the lower-level local access networks are of star-type. The problem is modeled as a mixed 01 integer programming, the special structure of which is exploited for the development of a dual-based lower bounding procedure. A heuristic procedure is developed to construct...

⎯In this paper, we construct a pair of Mond-Weir type second order symmetric dual problems, in which the objective function contains support function and is, therefore, nondifferentiable. For this pair of problems, we validate weak, strong and converse duality theorems under pseudobonvexity – pseudoboncavity assumption on the kernel function that appears in the problems. A second order self dua...