Convex hulls of monomials have been widely studied in the literature, and monomial convexifications are implemented in global optimization software for relaxing polynomials. However, there has been no study of the error in the global optimum from such approaches. We give bounds on the worst-case error for convexifying a monomial over subsets of [0, 1]. This implies additive error bounds for rel...

We know that the same set of selected extrema can be represented by a large number of different monotonicity graphs. We are interested in understanding how different, or how similar, the converged reconstruction results can be with respect to different monotonicity graphs. To do so, we made the following experiment. We constructed 100 random monotonicity graphs for the 2D vorticity data set (Fi...

The purpose of this paper is to describe three techniques that are useful for the investigation of monotone sets and multifunctions, and give two applications of them. The three techniques use the “fg–theorem”, the “big convexification” of a subset of E × E∗ or a multifunction E 7→ 2E (we suppose throughout that E is a nontrivial real Banach space with dual E∗), and the “convex function associa...

of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STRONG VALID INEQUALITIES FOR MIXED-INTEGER NONLINEAR PROGRAMS VIA DISJUNCTIVE PROGRAMMING AND LIFTING By Kwanghun Chung August 2010 Chair: Jean-Philippe. P. Richard Major: Industrial and Systems Engineering Mixed-Integer Nonlinear Progr...

This paper deals with the non-convex power system state estimation (PSSE) problem, which plays a central role in the monitoring and operation of electric power networks. Given a set of noisy measurements, PSSE aims at estimating the vector of complex voltages at all buses of the network. This is a challenging task due to the inherent nonlinearity of power flows, for which existing methods lack ...

In order for primal-dual methods to be applicable to a constrained minimization problem, it is necessary that restrictive convexity conditions are satisfied. In this paper, we consider a procedure by means of which a nonconvex problem is convexified and transformed into one which can be solved with the aid of primal-dual methods. Under this transformation, separability of the type necessary for...

In this paper we review the Disjunctive Decomposition (D) algorithm for two-stage Stochastic Mixed Integer Programs (SMIP). This novel method uses the principles of disjunctive programming to develop cuttingplane-based approximations of the feasible set of the second stage problem. At the core of this approach is the Common Cut Coefficient Theorem, which provides a mechanism for transforming cu...