Equivalences and differences in conic relaxations of combinatorial quadratic optimization problems

Introduction. The quadratic assignment problem is a well-established NP-hard problem in combinatorial optimization with applications in diverse fields like economics, archaeology, and chemistry. Due to its complexity, research on efficient solution methods remains active, including efforts for parallelization on multiprocessor computing systems. However, effective parallel algorithms are crucial to fully leverage these computational resources. In this context, algorithm unions (portfolios and teams) play a significant role in achieving parallel execution for solving such problems. Research objectives. This work investigates the application of portfolios constructed from modifications of the repeated iterated tabu search algorithm to the quadratic assignment problem. The effectiveness of these portfolios was evaluated through experimental computations. Results. The portfolios, derived from modifications of the repeated iterated tabu search algorithm, were applied to the quadratic assignment problem. For the most demanding test instances, the proposed algorithms were evaluated on the SCIT-4 supercomputer, alongside previously published results from the authors, confirming their competitive performance. Additionally, we assessed the parallel efficiency of these portfolios in solving instances of the quadratic assignment problem. The results demonstrate their ability to accelerate the optimization process (with speedup dependent on problem size and utilized processors), enabling the solution of large-scale problems. Conclusions. The conducted studies demonstrate that employing algorithm portfolios significantly accelerates the solution process for the quadratic assignment problem. Analysis of the results reveals a near-linear speedup factor achieved by the portfolio. For the challenging test instance tai100a, a new best solution value of 21040996 was obtained using a portfolio of 16 algorithms. Keywords: quadratic assignment problem, algorithm portfolios, experimental research, algorithm portfolios efficiency.

Differential evolution (DE) was introduced by Stone and Price in 1995 as a population-based stochastic search technique for solving optimization problems in a continuous space. DE has been successfully applied to various real world numerical optimization problems. In recent years not only continuous real-valued function, the applications of DE on combinatorial optimization problems with discrete decision variables are reported. However, genetic operator in the standard DE can not directly applied to discrete space. In this paper, we propose a method to solve quadratic assignment problems (QAP) by DE. The QAP is a well-known combinatorial optimization problem with a wide variety of practical applications. It is NP-hard and is considered to be one of the most difficult problems. In the QAP, a candidate solution can represented a permutation of integer. The proposed method employs permutation representation for individuals in DE. Therefore, a individual vector is encoded directly as a permutation. In discrete space, to realize efficient solution search like standard DE which have continuous nature, we modify differential operator to handle permutation encoding. Additionally, in order to maintain diversity of population, restart strategy and tabu list are introduced to proposed method instead of crossover operator. Finally, we show the experimental results using instances of QAPLIB and the efficacy of proposed method.

Recently, an interesting analogy between problems in combinatorial optimization and statistical mechanics has been developed and has proven useful in solving certain traditional optimization problems such as computer design, partitioning, component placement, wiring, and traveling salesman problems. The analogy has resulted in a methodology, termed "simulated annealing," which, in the process of iterating to an optimum, uses Monte Carlo sampling to occasionally accept solutions to discrete optimization problems which increase rather than decrease the objective function value. This process is counter to the normal 'steepest-descent' algorithmic approach. However, it is argued in the analogy that by taking such controlled uphill steps, the optimizing algorithm need not get "stuck" on inferior solutions. This paper presents an application of the simulated annealing method to solve the quadratic assignment problem (QAP). Performance is tested on a set of "standard" problems, as well as some newly generated larger problems (n = 50 and n = 100). The results are compared to those from other traditional heuristics, e.g., CRAFT, biased sampling, and a revised Hillier procedure. It is shown that under certain conditions simulated annealing can yield higher quality (lower cost) solutions at comparable CPU times. However, the simulated annealing algorithm is sensitive to a number of parameters, some of whose effects are investigated and reported herein through the analysis of an experimental design.

Knowledge representation of mathematical optimization problems and constructs for modeling

In this paper, we describe an implementation of the iterated tabu search (ITS) algorithm for the quadratic assignment problem (QAP), which is one of the well-known problems in combinatorial optimization. The medium- and large-scale QAPs are not, to this date, practically solvable to optimality, therefore heuristic algorithms are widely used. In the proposed ITS approach, intensification and diversification mechanisms are combined in a proper way. The goal of intensification is to search for good solutions in the neighbourhood of a given solution, while diversification is responsible for escaping from local optima and moving towards new regions of the search space. In particular, the following enhancements were implemented: new formula for fast evaluation of the objective function and efficient data structure; extended intensification mechanisms (including randomized tabu criterion, combination of tabu search and local search, dynamic tabu list maintaining); enhanced diversification strategy using periodic tabu tenure and special mutation procedure. The ITS algorithm is tested on the different instances taken from the QAP library QAPLIB. The results from the experiments demonstrate promising efficiency of the proposed algorithm, especially for the random QAP instances.

Many combinatorial optimization problems can be reduced to the LP problems using the results from the field called “polyhedral combinatorics”. The main goal of polyhedral combinatorics is to represent the convex envelope of feasible points of the given combinatorial problem in the form of a system of linear equalities and inequalities. Obtained by this way LP problems often have an exponentially growing number of constraints. We know that at each step of ellipsoid method for LP we have to use no more than one constraint from the set of constraints which are not fulfilled at current point. In many cases the problem of finding such a constraint can be formulated in the form of a new combinatorial (in some sense polar to the original) optimization problem (so-called separation problem).

The facility layout design problem is an extensively studied research problem and belongs to nonpolynomial hard (NP-hard) combinatorial optimization problem. Quadratic assignment problem (QAP) is one of the formulations that is investigated for facility layout design because of its wide applicability. Ant colony optimization (ACO), a biologically inspired heuristic has centered on solving the QAP by achieving approximation as good as possible. This paper presents a population-based hybrid ant system (PHAS), which is an extension of the hybrid ant system (HAS) in which the size of the ant colony has been fixed. The performance of the proposed ant algorithm for QAP is compared with the existing metaheuristic implementations such as tabu search, reactive tabu search, simulated annealing, genetic hybrid method, HAS, and max–min ant system. The experimental results show that the proposed PHAS perform significantly better than the other existing algorithms of QAP.

A review on the studies employing artificial bee colony algorithm to solve combinatorial optimization problems

Combinatorial optimization is one of the areas that has been studied intensively in computer science and operations research. Solving a Combinatorial Optimization Problem (COP) is NP-hard [55] and it involves finding a set of feasible solutions with the least-cost in a discrete search space [90]. It becomes complicated when the COP being solved is with large scale data, constraints and variables. The significance of COPs can be found in various types of industrial applications. Examples include finding an optimized scan chains route in integrated chips testing and job scheduling in manufacturing industries. This thesis presents a generic Bee Colony Optimization (BCO) framework for solving different COPs. The framework realizes computationally the collective intelligence shown in bee foraging behaviour. In a bee colony, bees are sent out of their beehive and travel across different locations to discover and to collect food. When a bee found a food source, it would perform waggle dance upon flying back to its hive. Its intension is to recruit more bees towards the newly discovered food source. Waggle dance operates as an important communication tool among bees. This collective intelligence shown in the bee foraging behaviour is modelled algorithmically as a generic framework to solve different COPs. The generic BCO framework is designed using the object-oriented approach. A set of domain independent and abstract classes are defined. The domain independent classes implement the bee foraging behaviour and the abstract classes act as an interface to interact with domain specific classes. To solve a COP with the proposed framework, the implementation of domain specific tasks as defined in the abstract classes needs to be supplied to the framework. Besides the advantage of solving different COPs, the proposed BCO framework can be modified or enhanced flexibly and the effect of the enhancement will be applicable across all different COPs being solved.

This paper studies the semidefinite programming SDP problem, i.e., the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First the classical cone duality is reviewed as it is specialized to SDP is reviewed. Next an interior point algorithm is presented that converges to the optimal solution in polynomial time. The approach is a direct extension of Ye’s projective method for linear programming. It is also argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion. Finally, the significance of these results is studied in a variety of combinatorial optimization problems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k-partite subgraph problem in graph...

Applying the attribute based hill climber heuristic to the vehicle routing problem

The ant colony optimization (ACO) is a metaheuristic, which has been successfully used to solve computationally difficult optimization problems, especially combinatorial optimization problems which belong to the class of non-deterministic polynomial (NP)-hard problems. This chapter explains ACO algorithms, their most important variants, and hybridization with local optimization methods. Practical considerations for successful ACO implementation and parameter setting are given. The working of the algorithm is explained in detail by using simple examples. The chapter ends with overview of research trends in ACO.

Contributors

Preface. 1 Introduction. 1.1 Two examples of combinatorial optimization. 1.2 Why study combinatorial optimization using statistical physics? 1.3 Textbooks. Bibliography. 2 Algorithms. 2.1 Pidgin Algol. 2.2 Iteration and recursion. 2.3 Divide-and-conquer. 2.4 Dynamic programming. 2.5 Backtracking. Bibliography. 3 Introduction to graphs. 3.1 Basic concepts and graph problems. 3.2 Basic graph algorithms. 3.3 Random graphs. Bibliography. 4 Introduction to complexity theory. 4.1 Turing machines. 4.2 Church's thesis. 4.3 Languages. 4.4 The halting problem. 4.5 Class P. 4.6 Class NP. 4.7 Definition of NP-completeness. 4.8 NP-complete problems. 4.9 Worst-case vs. typical-case complexity. Bibliography. 5 Statistical mechanics of the Ising model. 5.1 Phase transitions. 5.2 Some general notes on statistical mechanics. 5.3 The Curie-Weiss model of a ferromagnet. 5.4 The Ising model on a random graph. Bibliography. 6 Algorithms and numerical results for vertex covers. 6.1 Definitions. 6.2 Heuristic algorithms. 6.3 Branch-and-bound algorithm. 6.4 Results: Covering random graphs. 6.5 The leaf-removal algorithm. 6.6 Monte Carlo simulations. 6.7 Backbone. 6.8 Clustering of minimum vertex covers. Bibliography. 7 Statistical mechanics of vertex covers on a random graph. 7.1 Introduction. 7.2 The first-moment bound. 7.3 The hard-core lattice gas. 7.4 Replica approach. Bibliography. 8 The dynamics of vertex-cover algorithms. 8.1 The typical-case solution time of a complete algorithm. 8.2 The dynamics of generalized leaf-removal algorithms. 8.3 Random restart algorithms. Bibliography. 9 Towards new, statistical-mechanics motivated algorithms. 9.1 The cavity graph. 9.2 Warning propagation. 9.3 Belief propagation. 9.4 Survey propagation. 9.5 Numerical experiments on random graphs. Bibliography. 10 The satisfiability problem. 10.1 SAT algorithms. 10.2 Phase transitions in random K-SAT. 10.3 Typical-case dynamics of RandomWalkSAT. 10.4 Message-passing algorithms for SAT. Bibliography. 11 Optimization problems in physics. 11.1 Monte Carlo optimization. 11.2 Hysteric optimization. 11.3 Genetic algorithms. 11.4 Shortest paths and polymers in random media. 11.5 Maximum flows and random-field systems. 11.6 Submodular functions and free energy of Potts model. 11.7 Matchings and spin glasses. Bibliography. Index.

Parameter Selection of Discrete Particle Swarm Optimization Algorithm for the Quadratic Assignment Problems