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 $[0,1]^n$. This … Read more
A usual way to collect data in a Wireless Sensor Network (WSN) is by the support of a special agent, called data mule, that moves between sensor nodes and performs all communication between them. In this work, the focus is on the construction of the route that the data mule must follow to serve all … Read more
We study the convex hull of the intersection of a disjunctive set defined by parallel hyperplanes and the feasible set of a mixed integer second order cone optimization problem. We extend our prior work on disjunctive conic cuts, which has thus far been restricted to the case in which the intersection of the hyperplanes and … Read more
We study the trust region subproblem (TRS) of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints. Despite having a nonconvex objective, it is known that the TRS and a number of its variants are polynomial-time solvable. In this paper, we follow a second-order cone based approach to derive an exact … Read more
Let $\F$ be a quadratically constrained, possibly nonconvex, bounded set, and let $\E_1, \ldots, \E_l$ denote ellipsoids contained in $\F$ with non-intersecting interiors. We prove that minimizing an arbitrary quadratic $q(\cdot)$ over $\G := \F \setminus \cup_{k=1}^\ell \myint(\E_k)$ is no more difficult than minimizing $q(\cdot)$ over $\F$ in the following sense: if a given semidefinite-programming … Read more
Given a p-dimensional nonnegative, integral vector α, this paper characterizes the convex hull of the set S of nonnegative, integral vectors x that is lexicographically less than or equal to α. To obtain a finite number of elements in S, the vectors x are restricted to be component-wise upper-bounded by an integral vector u. We … Read more
We study two-stage stochastic mixed integer programs (TSS-MIPs) with integer variables in the second stage. We show that under suitable conditions, the second stage MIPs can be convexified by adding parametric cuts a priori. As special cases, we extend the results of Miller and Wolsey (Math Program 98(1):73-88, 2003) to TSS-MIPs. Furthermore, we consider second … Read more
This paper provides the convex hull description for the following basic operating constraints of a single power generation unit in Unit Commitment (UC) problems: 1) generation limits, 2) startup and shutdown capabilities, and 3) minimum up and down times. Although the model does not consider some crucial constraints, such as ramping, the proposed constraints can … Read more
This paper provides the convex hull description for the basic operation of slow- and quick-start units in power-based unit commitment (UC) problems. The basic operating constraints that are modeled for both types of units are: 1) generation limits and 2) minimum up and down times. Apart from this, the startup and shutdown processes are also … Read more
A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown—by several authors using different techniques—that the convex hull of the intersection of an ellipsoid, $\E$, and a split disjunction, $(l – x_j)(x_j – u) \le 0$ with $l < u$, equals the intersection ... Read more