We address combinatorial optimization problems with uncertain coefficients varying over ellipsoidal uncertainty sets. The robust counterpart of such a problem can be rewritten as a second-oder cone program (SOCP) with integrality constraints. We propose a branch-and-bound algorithm where dual bounds are computed by means of an active set algorithm. The latter is applied to the … Read more
This paper presents rigorous filtering methods for constraint satisfaction problems based on the interval union arithmetic. Interval unions are finite sets of closed and disjoint intervals that generalize the interval arithmetic. They allow a natural representation of the solution set of interval powers, trigonometric functions and the division by intervals containing zero. We show that … Read more
Let $r_{n}$ be the radius of the smallest circle into which one can pack $n$ non-overlapping unit squares that are free to rotate. For $n = 1, 2$, $r_{1} = \frac{\sqrt{2}}{2}$ and $r_{2} = \frac{\sqrt{5}}{2}$ are proved to be optimal. For $n \geq 3$ only guesses of $r_{n}$ are known. This paper introduces a computer-assisted … Read more
The software package BBCPOP is a MATLAB implementation of a hierarchy of sparse doubly nonnegative (DNN) relaxations of a class of polynomial optimization (minimization) problems (POPs) with binary, box and complementarity (BBC) constraints. Given a POP in the class and a relaxation order, BBCPOP constructs a simple conic optimization problem (COP), which serves as a … Read more
BBCPOP proposed in [4] is a MATLAB implementation of a hierarchy of sparse doubly nonnegative (DNN) relaxations of a class of polynomial optimization (minimization) problems (POPs) with binary, box and complementarity constraints. Given a POP in the class and a relaxation order (or a hierarchy level), BBCPOP constructs a simple conic optimization problem (COP), which … Read more
One of the relevant research topics to which Chris Floudas contributed was quadratically constrained quadratic programming (QCQP). This paper considers one of the simplest hard cases of QCQP, the two trust region subproblem (TTRS). In this case, one needs to minimize a quadratic function constrained by the intersection of two ellipsoids. The Lagrangian dual of … Read more
A bipartite bilinear program (BBP) is a quadratically constrained quadratic optimization problem where the variables can be partitioned into two sets such that fixing the variables in any one of the sets results in a linear program. We propose a new second order cone representable (SOCP) relaxation for BBP, which we show is stronger than … Read more
Discretization-based algorithms are proposed for the global solution of mixed-integer nonlinear generalized semi-infinite (GSIP) and bilevel (BLP) programs with lower-level equality constraints coupling the lower and upper level. The algorithms are extensions, respectively, of the algorithm proposed by Mitsos and Tsoukalas (J Glob Optim 61(1):1–17, 2015. https://doi.org/10.1007/s10898-014-0146-6) and by Mitsos (J Glob Optim 47(4):557–582, 2010. … Read more
We prove that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether the optimal value of a nonlinear optimization problem where the objective and constraints are given by low-degree polynomials is attained. If the degrees of these polynomials are fixed, our results along with previously-known “Frank-Wolfe type” theorems … Read more
Even in a static setting, the economic load dispatch problem (ELDP)—namely the cost-optimal distribution of power among generating units to meet a specific demand subject to system constraints—turns out to be a challenge owing to the consideration of valve-point effects (VPE), which make the cost function nonsmooth and nonconvex. We present a new method, termed … Read more