A Survey of Lagrangean Techniques for Discrete Optimization

In this paper, the matrix game based on triangular intuitionistic fuzzy payoff is put forward. Then, we get a conclusion that the equilibrium solution of this game model is equivalent to the solution of a pair of the primal–dual single objective intuitionistic fuzzy linear optimization problems ( I F L O P 1 ) and ( I F L O D 1 ) . Furthermore, by applying the accuracy function, which is linear, we transform the primal–dual single objective intuitionistic fuzzy linear optimization problems ( I F L O P 1 ) and ( I F L O D 1 ) into the primal–dual discrete linear optimization problems ( G L O P 1 ) and ( G L O D 1 ) . The above primal–dual pair ( G L O P 1 ) – ( G L O D 1 ) is symmetric in the sense the dual of ( G L O D 1 ) is ( G L O P 1 ) . Thus the primal–dual discrete linear optimization problems ( G L O P 1 ) and ( G L O D 1 ) are called the symmetric primal–dual discrete linear optimization problems. Finally, the technique is illustrated by an example.

Elementary Linear Programming with Applications

We consider in this paper the adaptation of heuristics used for programming nondeterministic games to the problems of discrete optimization. In particular, we use some “game” heuristic methods of decision-making in various discrete optimization problems. The object of each of these problems is programming anytime algorithms. Among the problems described in this paper, there are the classical traveling salesman problem and some connected problems of minimization for nondeterministic finite automata. The first of the considered methods is the geometrical approach to some discrete optimization problems. For this approach, we define some special characteristics relating to some initial particular case of considered discrete optimization problem. For instance, one of such statistical characteristics for the traveling salesman problem is a significant development of the so-called “distance functions” up to the geometric variant such problem. And using this distance, we choose the corresponding specific algorithms for solving the problem. Besides, other considered methods for solving these problems are constructed on the basis of special combination of some heuristics, which belong to some different areas of the theory of artificial intelligence. More precisely, we shall use some modifications of unfinished branchand-bound method; for the selecting immediate step using some heuristics, we apply dynamic risk functions; simultaneously for the selection of coefficients of the averaging-out, we also use genetic algorithms; and the reductive self-learning by the same genetic methods is also used for the start of unfinished branch-and-bound method again. This combination of heuristics represents a special approach to construction of anytime-algorithms for the discrete optimization problems. This approach can be considered as an alternative to application of methods of linear programming, and to methods of multi-agent optimization, and also to neural networks.

Population-based approaches have given us new search strategies and ideas in order to solve optimization problems. Usually, these methods are based on the performance carried out by a finite number of agents, which by the interaction between them they evolve and work all over the search space. Also, it is well-known that the correct employment of parameter values in this kind of method can positively impact their performance and behavior. In this context, the present work focuses on the design of a hybrid architecture which smartly balances the population size on run-time. In order to smartly balance and control the population size, a modular approach, named Linear Modular Population Balancer (LMPB), is proposed. The main ideas behind the designed architecture include the solving strategy behind a population-based metaheuristic, the influence of learning components based on multiple statistical modeling methods which transform the dynamic data generated into knowledge, and the possibilities to tackle both discrete and continuous optimization problems. In this regard, three modules are proposed for LMPB, which concern tasks such as the management of the population-based algorithm, parameter setting, probabilities, learning methods, and selection mechanism for the population size to employ. In order to test the viability and effectiveness of our proposed approach, we solve a set of well-known benchmark functions and the multidimensional knapsack problem (MKP). Additionally, we illustrate promising solving results, compare them against state-of-the-art methods which have proved to be good options for solving optimization problems, and give solid arguments for future work in the necessity to keep evolving this type of proposed architecture.

Organization of parallel calculations with the use of MPI functions in problems of discrete optimization is considered. The branch and bound method is applied to problems of integer linear and integer quadratic programming as well as to problems of set covering. The efficiency of the algorithms is analyzed on the basis of numerical experiments.

Contributors

Many discrete optimization problems can be formulated as either integer linear programming problems or constraint satisfaction problems. Although ILP methods appear to be more powerful, sometimes constraint programming can solve these problems more quickly. This paper describes a problem in which the difference in performance between the two approaches was particularly marked, since a solution could not be found using ILP.

CHAPTER 2.8 - Convexity and Discrete Optimization

Ranking in quadratic integer programming problems

An approach for the concurrent homogenization-based microstructure type and topology optimization problem

Simulation-based optimization using simulated annealing with ranking and selection

Simultaneous generalized hill climbing (SGHC) algorithms provide a framework for using heuristics to simultaneously address sets of intractable discrete optimization problems where information is shared between the problems during the algorithm execution. Many well-known heuristics can be embedded within the SGHC algorithm framework. This paper shows that the solutions generated by an SGHC algorithm are a stochastic process that satisfies the Markov property. This allows problem probability mass functions to be formulated for particular sets of problems based on the long-term behavior of the algorithm. Such results can be used to determine the proportion of iterations that an SGHC algorithm will spend optimizing over each discrete optimization problem. Sufficient conditions that guarantee that the algorithm spends an equal number of iterations in each discrete optimization problem are provided. SGHC algorithms can also be formulated such that the overall performance of the algorithm is independent of the initial discrete optimization problem chosen. Sufficient conditions are obtained guaranteeing that an SGHC algorithm will visit the globally optimal solution for each discrete optimization problem. Lastly, rates of convergence for SGHC algorithms are reported that show that given a rate of convergence for the embedded GHC algorithm, the SGHC algorithm can be designed to preserve this rate.

Contributors

Stochastic (mixed-) integer programs pose a great algorithmic and computational challenge in that they combine two generally difficult classes of problems: stochastic programs and discrete optimization problems. Exploring its dual angular structure, various decomposition methods have been widely studied, including Benders’ decomposition, Lagrangian relaxation, and test-set decomposition. These decomposition methods are often combined with search procedures such as branch-and-bound or branch-andcut. Within the confines of these broad frameworks, fine-tuned algorithms have been proposed to overcome obstacles such as nonconvexity of the secondstage value functions under integer recourse, and to take advantage of the many similar structured scenario subproblems using variable transformations. In this chapter, we survey some recent algorithms developed to solve twostage stochastic (mixed-) integer programs, as well as provide insights into and results concerning their interconnections, particularly for alternative convexification techniques.

In this paper, we propose a new hybrid social spider algorithm with simplex Nelder-Mead method in order to solve integer programming and minimax problems. We call the proposed algorithm a Simplex Social Spider optimization (SSSO) algorithm. In the the proposed SSSO algorithm, we combine the social spider algorithm with its powerful capability of performing exploration, exploitation, and the Nelder-Mead method in order to refine the best obtained solution from the standard social spider algorithm. In order to investigate the general performance of the proposed SSSO algorithm, we test it on 7 integer programming problems and 10 minimax problems and compare against 10 algorithms for solving integer programming problems and 9 algorithms for solving minimax problems. The experiments results show the efficiency of the proposed algorithm and its ability to solve integer and minimax optimization problems in reasonable time.