Convex envelopes of bounded monomials on two-variable cones

We consider an $n$-variate monomial function that is restricted both in value by lower and upper bounds and in domain by two homogeneous linear inequalities. Such functions are building blocks of several problems found in practical applications, and that fall under the class of Mixed Integer Nonlinear Optimization. We show that the upper envelope of the function in the given domain, for $n\ge 2$ is given by a conic inequality. We also present the lower envelope for $n=2$. To assess the applicability of branching rules based on homogeneous linear inequalities, we also derive the volume of the convex hull for $n=2$.Comment: 22 pages, 12 figure

Linear Programming Relaxations of Quadratically Constrained Quadratic Programs

We investigate the use of linear programming tools for solving semidefinite programming relaxations of quadratically constrained quadratic problems. Classes of valid linear inequalities are presented, including sparse PSD cuts, and principal minors PSD cuts. Computational results based on instances from the literature are presented.Comment: Published in IMA Volumes in Mathematics and its Applications, 2012, Volume 15

A local branching heuristic for MINLPs

Local branching is an improvement heuristic, developed within the context of branch-and-bound algorithms for MILPs, which has proved to be very effective in practice. For the binary case, it is based on defining a neighbourhood of the current incumbent solution by allowing only a few binary variables to flip their value, through the addition of a local branching constraint. The neighbourhood is then explored with a branch-and-bound solver. We propose a local branching scheme for (nonconvex) MINLPs which is based on iteratively solving MILPs and NLPs. Preliminary computational experiments show that this approach is able to improve the incumbent solution on the majority of the test instances, requiring only a short CPU time. Moreover, we provide algorithmic ideas for a primal heuristic whose purpose is to find a first feasible solution, based on the same scheme

Rounding-based heuristics for nonconvex MINLPs

We propose two primal heuristics for nonconvex mixed-integer nonlinear programs. Both are based on the idea of rounding the solution of a continuous nonlinear program subject to linear constraints. Each rounding step is accomplished through the solution of a mixed-integer linear program. Our heuristics use the same algorithmic scheme, but they differ in the choice of the point to be rounded (which is feasible for nonlinear constraints but possibly fractional) and in the linear constraints. We propose a feasibility heuristic, that aims at finding an initial feasible solution, and an improvement heuristic, whose purpose is to search for an improved solution within the neighborhood of a given point. The neighborhood is defined through local branching cuts or box constraints. Computational results show the effectiveness in practice of these simple ideas, implemented within an open-source solver for nonconvex mixed-integer nonlinear programs

A Randomized Algorithm for the MaxFS Problem

We consider the NP-hard combinatorial optimization problem of finding a feasible subsystem of maximum cardinality among a given set of linear inequalities. In some new and challenging applications in digital broadcasting and in the modelling of protein folding potentials one faces very large MaxFS instances with up to millions of inequalities in thousands of variables. We introduce and analyze a randomized algorithm that is surprisingly successful at solving MaxFS instances that arise in these contexts and exhibit some numerical results.\ud \ud Raphael Hauser was supported through grant NAL/00720/G from the Nuffield Foundation and through grant GR/M30975 from the Engineering and Physical Sciences Research Council of the U

Efficient Storage of Pareto Points in Biobjective Mixed Integer Programming

In biobjective mixed integer linear programs (BOMILPs), two linear objectives are minimized over a polyhedron while restricting some of the variables to be integer. Since many of the techniques for finding or approximating the Pareto set of a BOMILP use and update a subset of nondominated solutions, it is highly desirable to efficiently store this subset. We present a new data structure, a variant of a binary tree that takes as input points and line segments in $\R^2$ and stores the nondominated subset of this input. When used within an exact solution procedure, such as branch-and-bound (BB), at termination this structure contains the set of Pareto optimal solutions. We compare the efficiency of our structure in storing solutions to that of a dynamic list which updates via pairwise comparison. Then we use our data structure in two biobjective BB techniques available in the literature and solve three classes of instances of BOMILP, one of which is generated by us. The first experiment shows that our data structure handles up to $10^7$ points or segments much more efficiently than a dynamic list. The second experiment shows that our data structure handles points and segments much more efficiently than a list when used in a BB

On feasibility based bounds tightening

http://www.optimization-online.org/DB_HTML/2012/01/3325.htmlMathematical programming problems involving nonconvexities are usually solved to optimality using a (spatial) Branch-and-Bound algorithm. Algorithmic e?ciency depends on many factors, among which the widths of the bounding box for the problem variables at each Branch-and-Bound node naturally plays a critical role. The practically fastest box-tightening algorithm is known as FBBT (Feasibility-Based Bounds Tightening): an iterative procedure to tighten the variable ranges. Depending on the instance, FBBT may not converge ?nitely to its limit ranges, even in the case of linear constraints. Tolerance-based termination criteria yield ?nite termination, but not in worstcase polynomial-time. We model FBBT by using ?xed-point equations in terms of the variable bounding box, and we treat these equations as constraints of an auxiliary mathematical program. We demonstrate that the auxiliary mathematical problem is a linear program, which can of course be solved in polynomial time. We demonstrate the usefulness of our approach by improving an existing Branch-and-Bound implementation. global optimization, MINLP, spatial Branch-and-Bound, range reduction

On the convergence of feasibility based bounds tightening

Global Optimization and Mixed-Integer Nonlinear Programming problems such as min{f(x) | gL ≤ g(x) ≤ gU ∧ xL ≤ x ≤ xU ∧ ∀j ∈ Z (xj ∈ Z)}, where f : Rn → R, g : Rn → Rm, gL, gU ∈ Rm, xL, x, xU ∈ Rn and Z ⊆ {1, . . . , n},are usually solved to "-guaranteed approximation by the spatial Branch-and-Bound (sBB) algorithm [2], a variant of the usual Branch-and-Bound for dealing with nonlinear, possibly nonconvex f, g. Since the gap between the original problem P and its convex relaxation ¯ P is due both to integral variable restrictions being lifted as well as nonconvex functions being replaced by a convex relaxation, sBB is able to branch at continuous variables as well as integer ones. If ¯x solves ¯ P, the standard disjunction used at a node in the sBB search tree is xj ≤ ¯xj ∨xj ≥ ¯xj , the more usual one xj ≤ ⌊¯xj⌋∨xj ≥ ⌈¯xj⌉ being used only if j ∈ Z

Disjunctive Inequalities: Applications and Extensions

A general optimization problem can be expressed in the form min{cx: x ∈ S}, (1) where x ∈ R n is the vector of decision variables, c ∈ R n is a linear objective function and S ⊂ R n is the set of feasible solutions of (1). Because S is generall