Integer Programming Approach to HP Folding

One of the most widely studied protein structure prediction models is the hydrophobic-hydrophilic (HP) model, which explains the hydrophobic interaction and tries to maximize the number of contacts among hydrophobic amino-acids. In order to find a lower bound for the number of contacts, a number of heuristics have been proposed, but finding the optimal solution is still a challenge. In this research, we focus on creating a new integer programming model which is capable to provide tractable input for mixed-integer programming solvers, is general enough and allows relaxation with provable good upper bounds. Computational experiments using benchmark problems show that our formulation achieves these goals.

A Mixed Integer Quadratic Programming Model for the Low Autocorrelation Binary Sequence Problem

In this paper the low autocorrelation binary sequence problem (LABSP) is modeled as a mixed integer quadratic programming (MIQP) problem and proof of the model’s validity is given. Since the MIQP model is semidefinite, general optimization solvers can be used, and converge in a finite number of iterations. The experimental results show that IQP solvers, based on this MIQP formulation, are capable of optimally solving general/skew-symmetric LABSP instances of up to 30/51 elements in a moderate time. ACM Computing Classification System (1998): G.1.6, I.2.8.

CAPS in Z(2,n)

We consider point sets in (Z^2,n) where no three points are on a line – also called caps or arcs. For the determination of caps with maximum cardinality and complete caps with minimum cardinality we provide integer linear programming formulations and identify some values for small n.

Optimal Decision Rules in Logical Recognition Models

The task of smooth and stable decision rules construction in logical recognition models is considered. Logical regularities of classes are defined as conjunctions of one-place predicates that determine the membership of features values in an intervals of the real axis. The conjunctions are true on a special no extending subsets of reference objects of some class and are optimal. The standard approach of linear decision rules construction for given sets of logical regularities consists in realization of voting schemes. The weighting coefficients of voting procedures are done as heuristic ones or are as solutions of complex optimization task. The modifications of linear decision rules are proposed that are based on the search of maximal estimations of standard objects for their classes and use approximations of logical regularities by smooth sigmoid functions.

Mathematical Optimization for the Train Timetabling Problem

AMS Subj. Classification: 90C57; 90C10;

Solving Maximum Clique Problem for Protein Structure Similarity

Computing the similarity between two protein structures is a crucial task in molecular biology, and has been extensively investigated. Many protein structure comparison methods can be modeled as maximum weighted clique problems in specific k-partite graphs, referred here as alignment graphs. In this paper we present both a new integer programming formulation for solving such clique problems and a dedicated branch and bound algorithm for solving the maximum cardinality clique problem. Both approaches have been integrated in VAST, a software for aligning protein 3D structures largely used in the National Center for Biotechnology Information, an original clique solver which uses the well known Bron and Kerbosch algorithm (BK). Our computational results on real protein alignment instances show that our branch and bound algorithm is up to 116 times faster than BK.

A Fast Algorithm for Weber Problem on a Grid

AMS subject classification: 90B80.

Method of Linear Programming as an Idea for High School Curriculum

2010 Mathematics Subject Classification: 97D40, 97M10, 97M40, 97N60, 97N80, 97R80