A quantum-inspired Tabu search algorithm for solving combinatorial optimization problems

In this study, we propose a novel quantum-inspired evolutionary algorithm (QEA), called quantum inspired Tabu search (QTS). QTS is based on the classical Tabu search and characteristics of quantum computation, such as superposition. The process of qubit measurement is a probability operation that increases diversification; a quantum rotation gate used to searching toward attractive regions will increase intensification. This paper will show how to implement QTS into NP-complete problems such as 0/1 knapsack problems, multiple knapsack problems and the traveling salesman problem. These problems are important to computer science, cryptography and network security. Furthermore, our experimental results on 0/1 knapsack problems are compared with those of other heuristic algorithms, such as a conventional genetic algorithm, a Tabu search algorithm and the original QEA. The final outcomes show that QTS performs much better than other heuristic algorithms without premature convergence and with more efficiency. Also on multiple knapsack problems and the traveling salesman problem QTS verify its effectiveness.     

A new evolutionary search strategy for global optimization of high-dimensional problems

Global optimization of high-dimensional problems in practical applications remains a major challenge to the research community of evolutionary computation. The weakness of randomization-based evolutionary algorithms in searching high-dimensional spaces is demonstrated in this paper. A new strategy, SP-UCI is developed to treat complexity caused by high dimensionalities. This strategy features a slope-based searching kernel and a scheme of maintaining the particle population’s capability of searching over the full search space. Examinations of this strategy on a suite of sophisticated composition benchmark functions demonstrate that SP-UCI surpasses two popular algorithms, particle swarm optimizer (PSO) and differential evolution (DE), on high-dimensional problems. Experimental results also corroborate the argument that, in high-dimensional optimization, only problems with well-formative fitness landscapes are solvable, and slope-based schemes are preferable to randomization-based ones.     

On a sequential search strategy in global optimization problems

A class of sequential strategies for finding the global maximum of a real valued function is developed following a multistep optimality scheme. A new criterion is shown to give an optimal choice of each sequential step of the algorithm; numerical evidence is eventually reported which confirms the theoretical indications.     

An approximation algorithm for interval data minmax regret combinatorial optimization problems

The general problem of minimizing the maximal regret in combinatorial optimization problems with interval data is considered. In many cases, the minmax regret versions of the classical, polynomially solvable, combinatorial optimization problems become NP-hard and no approximation algorithms for them have been known. Our main result is a polynomial time approximation algorithm with a performance ratio of 2 for this class of problems.     

Bound sets for biobjective combinatorial optimization problems

In this paper we introduce the concept of bound sets for multiobjective discrete optimization. We prove general results on lower and upper bound sets for combinatorial optimization problems with multiple objectives. We present general algorithms for constructing lower and upper bound sets for biobjective problems and provide numerical results on five different problem types.     

Approximation algorithms for combinatorial multicriteria optimization problems

The computational complexity of combinatorial multiple objective programming problems is investigated. NP-completeness and # P -completeness results are presented. Using two definitions of approximability, general results are presented, which outline limits for approximation algorithms. The performance of the well-known tree and Christofides' heuristics for the traveling salesman problem is investigated in the multicriteria case with respect to the two definitions of approximability.     

Four-layer framework for combinatorial optimization problems domain

Four-layer framework for combinatorial optimization problems/models domain is suggested for applied problems structuring and solving: (1) basic combinatorial models and multicriteria decision making problems (e.g., clustering, knapsack problem, multiple choice problem, multicriteria ranking, assignment/allocation); (2) composite models/procedures (e.g., multicriteria combinatorial problems, morphological clique problem); (3) basic (standard) solving frameworks, e.g.: (i) Hierarchical Morphological Multicriteria Design (HMMD) (ranking, combinatorial synthesis based on morphological clique problem), (ii) multi-stage design (two-level HMMD), (iii) special multi-stage composite framework (clustering, assignment/location, multiple choice problem); and (4) domain-oriented solving frameworks, e.g.: (a) design of modular software, (b) design of test inputs for multi-function system testing, (c) combinatorial planning of medical treatment, (d) design and improvement of communication network topology, (e) multi-stage framework for information retrieval, (f) combinatorial evolution and forecasting of software, devices. The multi-layer approach covers ‘decision cycle’, i.e., problem statement, models, algorithms/procedures, solving schemes, decisions, decision analysis and improvement.     

Measuring instance difficulty for combinatorial optimization problems

Discovering the conditions under which an optimization algorithm or search heuristic will succeed or fail is critical for understanding the strengths and weaknesses of different algorithms, and for automated algorithm selection. Large scale experimental studies - studying the performance of a variety of optimization algorithms across a large collection of diverse problem instances - provide the resources to derive these conditions. Data mining techniques can be used to learn the relationships between the critical features of the instances and the performance of algorithms. This paper discusses how we can adequately characterize the features of a problem instance that have impact on difficulty in terms of algorithmic performance, and how such features can be defined and measured for various optimization problems. We provide a comprehensive survey of the research field with a focus on six combinatorial optimization problems: assignment, traveling salesman, and knapsack problems, bin-packing, graph coloring, and timetabling. For these problems - which are important abstractions of many real-world problems - we review hardness-revealing features as developed over decades of research, and we discuss the suitability of more problem-independent landscape metrics. We discuss how the features developed for one problem may be transferred to study related problems exhibiting similar structures.     

A negotiation framework for linked combinatorial optimization problems

We tackle the challenge of applying automated negotiation to self-interested agents with local but linked combinatorial optimization problems. Using a distributed production scheduling problem, we propose two negotiation strategies for making concessions in a joint search space of agreements. In the first strategy, building on Lai and Sycara (Group Decis Negot 18(2):169–187, 2009), an agent concedes on local utility in order to achieve an agreement. In the second strategy, an agent concedes on the distance in an attribute space while maximizing its local utility. Lastly, we introduce a Pareto improvement phase to bring the final agreement closer to the Pareto frontier. Experimental results show that the new attribute-space negotiation strategy outperforms its utility-based counterpart on the quality of the agreements and the Pareto improvement phase is effective in approaching the Pareto frontier. This article presents the first study of applying automated negotiation to self-interested agents each with a local, but linked, combinatorial optimization problem.     

An adaptive hybrid single-loop method for reliability-based design optimization using iterative control strategy

Single-loop approach (SLA) is one of the most promising methods for solving linear and weakly nonlinear reliability-based design optimization (RBDO) problems. However, since SLA locates the current approximate most probable point (MPP) by using the gradient information of the previous one to reduce the computational cost, it may lead to inaccuracy when the nonlinearity of probabilistic constraints becomes relatively high. To overcome this limitation, a new adaptive hybrid single-loop method (AH-SLM) that can automatically choose to search for the approximate MPP or accurate MPP is proposed in this paper. Moreover, to get the accurate MPP more efficiently and alleviate the oscillation in the search process, an iterative control strategy (ICS) with two iterative control criteria is developed. In each iterative step, the KKT-condition of performance measure approach (PMA) is introduced to check the validity of the approximate MPP. If the approximate MPP is infeasible, ICS will be further carried out to search for the accurate MPP. The two iterative control criteria are used to update the oscillation control step length, then ICS can converge fast for both weakly and highly nonlinear performance functions. Besides, numerical examples are presented to verify the efficiency and robustness of our proposed method.     

Continuous reductions among combinatorial optimization problems

Summary Many reductions among combinatorial problems are known in the context of NP-completeness. These reductions preserve the optimality of solutions. However, they may change the relative error of approximative solutions dramatically. In this paper, we apply a new type of reductions, called continuous reductions. When one problem is continuously reduced to another, any approximation algorithm for the latter problem can be transformed into an approximation algorithm for the former. Moreover, the performance ratio is preserved up to a constant factor. We relate the problem Minimum Number of Inverters in CMOS-Circuits, which arises in the context of logic synthesis, to several classical combinatorial problems such as Maximum Independent Set and Deletion of a Minimum Number of Vertices (Edges) in Order to Obtain a Bipartite (Partial) Subgraph.     

Probability Collectives: A multi-agent approach for solving combinatorial optimization problems

Complex systems generally have many components. It is not possible to understand such complex systems only by knowing the individual components and their behavior. This is because any move by a component affects the further decisions/moves by other components and so on. In a complex system, as the number of components grows, complexity also grows exponentially, making the entire system to be seen as a collection of subsystems or a Multi-Agent System (MAS). The major challenge is to make these agents work in a coordinated way, optimizing their local utilities and contributing the maximum towards optimization of the global objective. This paper discusses the theory of Collective Intelligence (COIN) using the modified version of Probability Collectives (PC) to achieve the global goal. The paper successfully demonstrated this approach by optimizing the Rosenbrock function in which the coupled variables are seen as autonomous agents working collectively to achieve the function optimum. To demonstrate the PC approach on combinatorial optimization problems, two test cases of the Multi-Depot Multiple Traveling Salesmen Problem (MDMTSP) with 3 depots, 3 vehicles and 15 nodes are solved. In these cases, the vehicles are considered as autonomous agents collectively searching the minimum cost path. PC is successfully accompanied with insertion, elimination and swapping heuristic techniques. The optimum results to the Rosenbrock function and both the MDMTSP test cases are obtained at reasonable computational costs.     

Improved cuckoo search for reliability optimization problems

An efficient approach to solve engineering optimization problems is the cuckoo search algorithm. It is a recently developed meta-heuristic optimization algorithm. Normally, the parameters of the cuckoo search are kept constant. This may result in decreasing the efficiency of the algorithm. To cope with this issue, the cuckoo search parameters should be tuned properly. In this paper, an improved cuckoo search algorithm, enhancing the accuracy and convergence rate of the cuckoo search algorithm, is presented. Then, the performance of the proposed algorithm is tested on some complex engineering optimization problems. They are four well-known reliability optimization problems, a large-scale reliability optimization problem as well as a complex system, which is a 15-unit system reliability optimization problem. Finally, the results are compared with those given by several well-known methods. Simulation results demonstrate the effectiveness of the proposed algorithm.     

A note on robustness tolerances for combinatorial optimization problems

We consider so-called generic combinatorial optimization problem, where the set of feasible solutions is some family of nonempty subsets of a finite ground set with specified positive initial weights of elements, and the objective function represents the total weight of elements of the feasible solution. We assume that the set of feasible solutions is fixed, but the weights of elements may be perturbed or are given with errors. All possible realizations of weights form the set of scenarios.A feasible solution, which for a given set of scenarios guarantees the minimum value of the worst-case relative regret among all the feasible solutions, is called a robust solution. The maximum percentage perturbation of a single weight, which does not destroy the robustness of a given solution, is called the robustness tolerance of this weight with respect to the solution considered.In this paper we give formulae for computing the robustness tolerances with respect to an optimal solution obtained for some initial weights and we show that this can be done in polynomial time whenever the optimization problem is polynomially solvable itself.     

A columnar competitive model for solving combinatorial optimization problems

The major drawbacks of the Hopfield network when it is applied to some combinatorial problems, e.g., the traveling salesman problem (TSP), are invalidity of the obtained solutions, trial-and-error setting value process of the network parameters and low-computation efficiency. This letter presents a columnar competitive model (CCM) which incorporates winner-takes-all (WTA) learning rule for solving the TSP. Theoretical analysis for the convergence of the CCM shows that the competitive computational neural network guarantees the convergence to valid states and avoids the onerous procedures of determining the penalty parameters. In addition, its intrinsic competitive learning mechanism enables a fast and effective evolving of the network. The simulation results illustrate that the competitive model offers more and better valid solutions as compared to the original Hopfield network.     

求解组合优化问题的改进型量子进化算法

根据组合优化问题的特点,提出了一种求解组合优化问题的改进型量子进化算法.借鉴小生境协同进化思想初始化种群,增加了个体多样性;采用动态策略调整量子门旋转角,加快了收敛速度;采用优体交叉策略实施染色体交叉操作,增强了局部搜索能力.利用典型组合优化问题--2个多维0/1背包问题实例对算法性能进行验证,结果表明了该算法的可行性和有效性.     

AND/OR Branch-and-Bound search for combinatorial optimization in graphical models

This is the first of two papers presenting and evaluating the power of a new framework for combinatorial optimization in graphical models, based on AND/OR search spaces. We introduce a new generation of depth-first Branch-and-Bound algorithms that explore the AND/OR search tree using static and dynamic variable orderings. The virtue of the AND/OR representation of the search space is that its size may be far smaller than that of a traditional OR representation, which can translate into significant time savings for search algorithms. The focus of this paper is on linear space search which explores the AND/OR search tree. In the second paper we explore memory intensive AND/OR search algorithms. In conjunction with the AND/OR search space we investigate the power of the mini-bucket heuristics in both static and dynamic setups. We focus on two most common optimization problems in graphical models: finding the Most Probable Explanation in Bayesian networks and solving Weighted CSPs. In extensive empirical evaluations we demonstrate that the new AND/OR Branch-and-Bound approach improves considerably over the traditional OR search strategy and show how various variable ordering schemes impact the performance of the AND/OR search scheme.     

Minmax regret combinatorial optimization problems with investments

A new minmax regret optimization model in a system with uncertain parameters is proposed. In this model it is allowed to make investments before a minmax regret solution is implemented in order to modify the source or the nature of the existing uncertainty. Therefore, it is allowed to spend resources in order to change the basic cost structure of the system and take advantage of the modified system to find a robust solution. Some properties of this model allow us to have proper Mathematical Programming formulations that can be solved by standard optimization packages. As a practical application we consider the shortest path problem in a network in which it is possible to modify the uncertainty intervals for the arc costs by investing in the system. We also give an approximate algorithm and generalize some existing results on constant factor approximations.