13 research outputs found
New bounds for truthful scheduling on two unrelated selfish machines
We consider the minimum makespan problem for tasks and two unrelated
parallel selfish machines. Let be the best approximation ratio of
randomized monotone scale-free algorithms. This class contains the most
efficient algorithms known for truthful scheduling on two machines. We propose
a new formulation for , as well as upper and lower bounds on
based on this formulation. For the lower bound, we exploit pointwise
approximations of cumulative distribution functions (CDFs). For the upper
bound, we construct randomized algorithms using distributions with piecewise
rational CDFs. Our method improves upon the existing bounds on for small
. In particular, we obtain almost tight bounds for showing that
.Comment: 28 pages, 3 tables, 1 figure. Theory Comput Syst (2019
Positive semidefinite approximations to the cone of copositive kernels
It has been shown that the maximum stable set problem in some infinite
graphs, and the kissing number problem in particular, reduces to a minimization
problem over the cone of copositive kernels. Optimizing over this infinite
dimensional cone is not tractable, and approximations of this cone have been
hardly considered in literature. We propose two convergent hierarchies of
subsets of copositive kernels, in terms of non-negative and positive definite
kernels. We use these hierarchies and representation theorems for invariant
positive definite kernels on the sphere to construct new SDP-based bounds on
the kissing number. This results in fast-to-compute upper bounds on the kissing
number that lie between the currently existing LP and SDP bounds.Comment: 29 pages, 2 tables, 1 figur
Adjustable Robust Two-Stage Polynomial Optimization with Application to AC Optimal Power Flow
In this work, we consider two-stage polynomial optimization problems under
uncertainty. In the first stage, one needs to decide upon the values of a
subset of optimization variables (control variables). In the second stage, the
uncertainty is revealed and the rest of optimization variables (state
variables) are set up as a solution to a known system of possibly non-linear
equations. This type of problem occurs, for instance, in optimization for
dynamical systems, such as electric power systems. We combine tools from
polynomial and robust optimization to provide a framework for general
adjustable robust polynomial optimization problems. In particular, we propose
an iterative algorithm to build a sequence of (approximately) robustly feasible
solutions with an improving objective value and verify robust feasibility or
infeasibility of the resulting solution under a semialgebraic uncertainty set.
At each iteration, the algorithm optimizes over a subset of the feasible set
and uses affine approximations of the second-stage equations while preserving
the non-linearity of other constraints. The algorithm allows for additional
simplifications in case of possibly non-convex quadratic problems under
ellipsoidal uncertainty. We implement our approach for AC Optimal Power Flow
and demonstrate the performance of our proposed method on Matpower instances.Comment: 28 pages, 3 table
Copositive certificates of non-negativity for polynomials on semialgebraic sets
A certificate of non-negativity is a way to write a given function so that
its non-negativity becomes evident. Certificates of non-negativity are
fundamental tools in optimization, and they underlie powerful algorithmic
techniques for various types of optimization problems. We propose certificates
of non-negativity of polynomials based on copositive polynomials. The
certificates we obtain are valid for generic semialgebraic sets and have a
fixed small degree, while commonly used sums-of-squares (SOS) certificates are
guaranteed to be valid only for compact semialgebraic sets and could have large
degree. Optimization over the cone of copositive polynomials is not tractable,
but this cone has been well studied. The main benefit of our copositive
certificates of non-negativity is their ability to translate results known
exclusively for copositive polynomials to more general semialgebraic sets. In
particular, we show how to use copositive polynomials to construct structured
(e.g., sparse) certificates of non-negativity, even for unstructured
semialgebraic sets. Last but not least, copositive certificates can be used to
obtain not only hierarchies of tractable lower bounds, but also hierarchies of
tractable upper bounds for polynomial optimization problems.Comment: 27 pages, 1 figur
The maximum -colorable subgraph problem and related problems
The maximum -colorable subgraph (MCS) problem is to find an induced
-colorable subgraph with maximum cardinality in a given graph. This paper is
an in-depth analysis of the MCS problem that considers various semidefinite
programming relaxations including their theoretical and numerical comparisons.
To simplify these relaxations we exploit the symmetry arising from permuting
the colors, as well as the symmetry of the given graphs when applicable. We
also show how to exploit invariance under permutations of the subsets for other
partition problems and how to use the MCS problem to derive bounds on the
chromatic number of a graph.
Our numerical results verify that the proposed relaxations provide strong
bounds for the MCS problem, and that those outperform existing bounds for
most of the test instances
The many faces of positivity to approximate structured optimization problems
The PhD dissertation proposes tractable linear and semidefinite relaxations for optimization problems that are hard to solve and approximate, such as polynomial or copositive problems. To do this, we exploit the structure and inherent symmetry of these problems. The thesis consists of five essays devoted to distinct problems. First, we consider the kissing number problem. The kissing number is the maximum number of non-overlapping unit spheres that can simultaneously touch another unit sphere, in n-dimensional space. In chapter two we construct a new hierarchy of upper bounds on the kissing number. To implement the hierarchy, in chapter three we propose two generalizations of Schoenberg's theorem on positive definite kernels. In the fourth chapter, we derive new certificates of non-negativity of polynomials on generic sets defined by polynomial equalities and inequalities. These certificates are based on copositive polynomials and allow obtaining new upper and lower bounds for polynomial optimization problems. In chapter five, for any given graph we look for the largest k-colorable subgraph; that is, the induced subgraph that can be colored in k colors such that no two adjacent vertices have the same color. We obtain several new semidefinite programming relaxations to this problem. In the final sixth chapter, we consider the problem of allocating tasks to unrelated parallel selfish machines to minimize the time to complete all the tasks. For this problem, we suggest new upper and lower bounds on the best approximation ratio of a class of truthful task allocation algorithms
Approximating the cone of copositive kernels to estimate the stability number of infinite graphs
It has been shown that the stable set problem in an infinite compact graph, and particularly the kissing number problem, reduces to an optimization problem over the cone of copositive kernels. We propose two converging hierarchies approximating this cone. Both are extensions of existing inner hierarchies for the finite dimensional copositive cone. We implement the first two levels of the new hierarchies for the kissing number problem