A parametric integer programming algorithm for bilevel mixed integer programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.Comment: 11 page

Compact relaxations for polynomial programming problems

Reduced RLT constraints are a special class of Reformulation- Linearization Technique (RLT) constraints. They apply to nonconvex (both continuous and mixed-integer) quadratic programming problems subject to systems of linear equality constraints. We present an extension to the general case of polynomial programming problems and discuss the derived convex relaxation. We then show how to perform rRLT constraint generation so as to reduce the number of inequality constraints in the relaxation, thereby making it more compact and faster to solve. We present some computational results validating our approach

On the composition of convex envelopes for quadrilinear terms

International audienceWithin the framework of the spatial Branch-and-Bound algorithm for solving Mixed-Integer Nonlinear Programs, different convex relaxations can be obtained for multilinear terms by applying associativity in different ways. The two groupings ((x1x2)x3)x4 and (x1x2x3)x4 of a quadrilinear term, for example, give rise to two different convex relaxations. In [6] we prove that having fewer groupings of longer terms yields tighter convex relaxations. In this paper we give an alternative proof of the same fact and perform a computational study to assess the impact of the tightened convex relaxation in a spatial Branch-and-Bound setting

An FPTAS for optimizing a class of low-rank functions over a polytope

We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of non-linear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Pareto-optimal front of the linear functions which constitute the given low-rank function. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasi-concavity on the objective function. For the special case of quasi-concave function minimization, we give an alternative FPTAS, which always returns a solution which is an extreme point of the polytope. Our technique can also be used to obtain an FPTAS for combinatorial optimization problems with non-linear objective functions, for example when the objective is a product of a fixed number of linear functions. We also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor, unless P = NP. We prove this by showing a similar hardness of approximation result for supermodular function minimization, a result that may be of independent interest

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Error bounds for monomial convexification in polynomial optimization

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 implies additive error bounds for relaxing a polynomial optimization problem by convexifying each monomial separately. Our main error bounds depend primarily on the degree of the monomial, making them easy to compute. Since monomial convexification studies depend on the bounds on the associated variables, in the second part, we conduct an error analysis for a multilinear monomial over two different types of box constraints. As part of this analysis, we also derive the convex hull of a multilinear monomial over $[-1,1]^n$.Comment: 33 pages, 2 figures, to appear in journa

Antimicrobial activity against oral pathogens and immunomodulatory effects and toxicity of geopropolis produced by the stingless bee Melipona fasciculata Smith

<p>Abstract</p> <p>Background</p> <p>Native bees of the tribe Meliponini produce a distinct kind of propolis called geopropolis. Although many pharmacological activities of propolis have already been demonstrated, little is known about geopropolis, particularly regarding its antimicrobial activity against oral pathogens. The present study aimed at investigating the antimicrobial activity of <it>M. fasciculata </it>geopropolis against oral pathogens, its effects on <it>S. mutans </it>biofilms, and the chemical contents of the extracts. A gel prepared with a geopropolis extract was also analyzed for its activity on <it>S. mutans </it>and its immunotoxicological potential.</p> <p>Methods</p> <p>Antimicrobial activities of three hydroalcoholic extracts (HAEs) of geopropolis, and hexane and chloroform fractions of one extract, were evaluated using the agar diffusion method and the broth dilution technique. Ethanol (70%, v/v) and chlorhexidine (0.12%, w/w) were used as negative and positive controls, respectively. Total phenol and flavonoid concentrations were assayed by spectrophotometry. Immunotoxicity was evaluated in mice by topical application in the oral cavity followed by quantification of biochemical and immunological parameters, and macro-microscopic analysis of animal organs.</p> <p>Results</p> <p>Two extracts, HAE-2 and HAE-3, showed inhibition zones ranging from 9 to 13 mm in diameter for <it>S. mutans </it>and <it>C. albicans</it>, but presented no activity against <it>L</it>. <it>acidophilus</it>. The MBCs for HAE-2 and HAE-3 against <it>S. mutans </it>were 6.25 mg/mL and 12.5 mg/mL, respectively. HAE-2 was fractionated, and its chloroform fraction had an MBC of 14.57 mg/mL. HAE-2 also exhibited bactericidal effects on <it>S. mutans </it>biofilms after 3 h of treatment. Significant differences (p < 0.05) in total phenol and flavonoid concentrations were observed among the samples. Signs toxic effects were not observed after application of the geopropolis-based gel, but an increase in the production of IL-4 and IL-10, anti-inflammatory cytokines, was detected.</p> <p>Conclusions</p> <p>In summary, geopropolis produced by <it>M. fasciculata </it>can exert antimicrobial action against <it>S. mutans </it>and <it>C. albicans</it>, with significant inhibitory activity against <it>S. mutans </it>biofilms. The extract with the highest flavonoid concentration, HAE-2, presented the highest antimicrobial activity. In addition, a geopropolis-based gel is not toxic in an animal model and displays anti-inflammatory effect.</p