49 research outputs found
Lower bounds on the size of semidefinite programming relaxations
We introduce a method for proving lower bounds on the efficacy of
semidefinite programming (SDP) relaxations for combinatorial problems. In
particular, we show that the cut, TSP, and stable set polytopes on -vertex
graphs are not the linear image of the feasible region of any SDP (i.e., any
spectrahedron) of dimension less than , for some constant .
This result yields the first super-polynomial lower bounds on the semidefinite
extension complexity of any explicit family of polytopes.
Our results follow from a general technique for proving lower bounds on the
positive semidefinite rank of a matrix. To this end, we establish a close
connection between arbitrary SDPs and those arising from the sum-of-squares SDP
hierarchy. For approximating maximum constraint satisfaction problems, we prove
that SDPs of polynomial-size are equivalent in power to those arising from
degree- sum-of-squares relaxations. This result implies, for instance,
that no family of polynomial-size SDP relaxations can achieve better than a
7/8-approximation for MAX-3-SAT
A Lagrangian relaxation approach to the edge-weighted clique problem
The -clique polytope is the convex hull of the node and edge incidence vectors of all subcliques of size at most of a complete graph on nodes. Including the Boolean quadric polytope as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over . The problem of optimizing linear functions over has so far been approached via heuristic combinatorial algorithms and cutting-plane methods. We study the structure of in further detail and present a new computational approach to the linear optimization problem based on Lucena's suggestion of integrating cutting planes into a Lagrangian relaxation of an integer programming problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented. \u
Integrality of Linearizations of Polynomials over Binary Variables using Additional Monomials
Polynomial optimization problems over binary variables can be expressed as
integer programs using a linearization with extra monomials in addition to
those arising in the given polynomial. We characterize when such a
linearization yields an integral relaxation polytope, generalizing work by Del
Pia and Khajavirad (SIAM Journal on Optimization, 2018) and Buchheim, Crama and
Rodr\'iguez-Heck (European Journal of Operations Research, 2019). We also
present an algorithm that finds these extra monomials for a given polynomial to
yield an integral relaxation polytope or determines that no such set of extra
monomials exists. In the former case, our approach yields an algorithm to solve
the given polynomial optimization problem as a compact LP, and we complement
this with a purely combinatorial algorithm.Comment: 27 pages, 11 figure
Tight Polyhedral Representations of Discrete Sets Using Projections, Simplices, and Base-2 Expansions
This research effort focuses on the acquisition of polyhedral outer-approximations to the convex hull of feasible solutions for mixed-integer linear and mixed-integer nonlinear programs. The goal is to produce desirable formulations that have superior size and/or relaxation strength. These two qualities often have great influence on the success of underlying solution strategies, and so it is with these qualities in mind that the work of this dissertation presents three distinct contributions. The first studies a family of relatively unknown polytopes that enable the linearization of polynomial expressions involving two discrete variables. Projections of higher-dimensional convex hulls are employed to reduce the dimensionality of the requisite linearizing polyhedra. For certain lower dimensions, a complete characterization of the convex hull is obtained; for others, a family of facets is acquired. Furthermore, a novel linearization for the product of a bounded continuous variable and a general discrete variable is obtained. The second contribution investigates the use of simplicial facets in the formation of novel convex hull representations for a class of mixed-discrete problems having a subset of their variables taking on discrete, affinely independent realizations. These simplicial facets provide new theoretical machinery necessary to extend the reformulation-linearization technique (RLT) for mixed-binary and mixed-discrete programs. In doing so, new insight is provided which allows for the subsumation of previous mixed-binary and mixed-discrete RLT results. The third contribution presents a novel approach for representing functions of discrete variables and their products using logarithmic numbers of 0-1 variables in order to economize on the number of these binary variables. Here, base-2 expansions are used within linear restrictions to enforce the appropriate behavior of functions of discrete variables. Products amongst functions are handled by scaling these linear restrictions. This approach provides insight into, improves upon, and subsumes recent related linearization methods from the literature