Hybrid Multi-Objective Combinatorial Optimization Technique with Improved Compatibility between GA and Gradient-Based Local Search

On an exact penalty function method for nonlinear mixed discrete programming problems and its applications in search engine advertising problems

In this paper we consider a particular class of nonlinear optimization problems involving both continuous and discrete variables. The distinguishing feature of this class of nonlinear mixed variable optimization problems is that the structure and the number of variables of the problem depend on the values of some discrete variables. In particular, we define a general algorithm model for the solution of this class of problems, that draws inspiration from the approach recently proposed by Audet and Dennis [SIAM J. Optim., 11 (2001), pp. 573--594], and is based on the strategy of combining in a suitable way a local search with respect to the continuous variables and a local search with respect to the discrete variables. We prove global convergence of the algorithm model without specifying the local continuous search, but only identifying some reasonable requirements. Moreover, we define a particular derivative-free algorithm for solving mixed variable programming problems where the continuous variables are linearly constrained and derivative information is not available. Finally, we report numerical results obtained by the proposed algorithm in solving a real optimal design problem. These results show the effectiveness of the approach.

In this paper, we consider a general class of nonlinear mixed discrete programming problems. By introducing continuous variables to replace the discrete variables, the problem is first transformed into an equivalent nonlinear continuous optimization problem subject to original constraints and additional linear and quadratic constraints. Then, an exact penalty function is employed to construct a sequence of unconstrained optimization problems, each of which can be solved effectively by unconstrained optimization techniques, such as conjugate gradient or quasi-Newton methods. It is shown that any local optimal solution of the unconstrained optimization problem is a local optimal solution of the transformed nonlinear constrained continuous optimization problem when the penalty parameter is sufficiently large. Numerical experiments are carried out to test the efficiency of the proposed method.

This chapter is devoted to the parametrization of memetic algorithms and how to find a good balance between global and local search. This is one of the most pressing questions when designing a hybrid algorithm. The idea of hybridization is to combine the advantages of different components. But if one components dominates another one, hybridization may become more hindering than useful and computational effort may be wasted. For the case of memetic algorithms, if the effect of local search is too strong, the algorithm may quickly get stuck in local optima of bad quality.Moreover, the algorithm is likely to rediscover the same local optimum over and over again. Lastly, an excessive local search quickly leads to a loss of diversity within the population.

Global and local game tree search

Differential evolution (DE) has been used to solve real-parameter optimization problems with nonlinear and multimodal functions for more than a decade of years. However, it is pointed out that this classical DE harbors restricted efficiency and limited local search ability. Inspired by that gradient-based algorithms have powerful local search ability, we propose a new algorithm, which is diversity-maintained DE based on gradient local search (namely, DMGBDE), by incorporating approximate gradient-based algorithms into the DE search while maintaining the diversity of the population. The primary novelties of the proposed DMGBDE are the following: (1) the gradient-based algorithm is embedded into DE in a different manner and (2) a diversity-maintained mutation is introduced to slow down the learning procedure from the searched best individual. We conduct numerical experiments with a number of benchmark problems to measure the performance of the proposed DMGBDE. Simulation results show that the proposed DMGBDE outperforms classical DE and variant without gradient local search or diversity-based mutation. Moreover, comparison with some other recently reported approaches indicates that our proposed DMGBDE is rather competitive.

In this paper, we present a multi-objective fractional probabilistic programming problem where the parameters in the objective functions and right-hand side constraints are multi-choice in nature. Multi-choice multi-objective objective fractional probabilistic programming problem is formulated. The multi-choice parameters in the right-hand side of constraints are random variables following Cauchy distribution. Multi-choice multi-objective fractional probabilistic programming problem is transformed to deterministic equivalent multi-choice multi-objective fractional programming problem using chance-constrained programming problem. Lagrange interpolating polynomial approach is proposed to deal with multi-choice parameters. The multi-choice multi-objective fractional programming problem is transformed to non-linear mixed integer multi-objective programming problem. An \(\epsilon -\) constrained method is applied to find the compromise solutions of non-linear mixed integer multi-objective programming problem using different values of \(\epsilon \). LINGO software is applied to find the optimal solution of single objective non-linear mixed integer programming problem. Finally, a numerical example is given to illustrate the proposed programming problem.KeywordsMulti-choice programming problemMulti-objective programming problemProbabilistic programming problemFractional programming problem\(\epsilon \)-constraint method

An important implementation issue in the design of hybrid evolutionary multiobjective optimization algorithms with local search (i.e., multiobjective memetic algorithms) is how to strike a balance between local search and global search. If local search is applied to all individuals at every generation, almost all computation time is spent by local search. As a result, global search ability of memetic algorithms is not well utilized. We can use three ideas for decreasing the computation load of local search. One idea is to apply local search to only a small number of individuals. This idea can be implemented by introducing a local search probability, which is used to choose only a small number of initial solutions for local search from the current population. Another idea is a periodical (i.e., intermittent) use of local search. This idea can be implemented by introducing a local search interval (e.g., every 10 generations), which is used to specify when local search is applied. The other idea is an early termination of local search. Local search for each initial solution is terminated after a small number of neighbors are examined. This idea can be implemented by introducing a local search length, which is the number of examined neighbors in a series of iterated local search from a single initial solution. In this paper, we discuss the use of these three ideas to strike a local-global search balance. Through computational experiments on a two-objective 500-item knapsack problem, we compare various settings of local search such as short local search from all individuals at every generation, long local search from only a few individuals at every generation, and periodical long local search from all individuals. Global search in this paper means genetic search by crossover and mutation in multiobjective memetic algorithms.

A hybrid global–local approach for optimization of injection gate locations in liquid composite molding process simulations

Nonlinear optimization algorithms are rarely discussed from a complexity point of view. Even the concept of solving nonlinear problems on digital computers is not well defined. The focus here is on a complexity approach for designing and analyzing algorithms for nonlinear optimization problems providing optimal solutions with prespecified accuracy in the solution space. We delineate the complexity status of convex problems over network constraints, dual of flow constraints, dual of multi-commodity, constraints defined by a submodular rank function (a generalized allocation problem), tree networks, diagonal dominant matrices, and nonlinear Knapsack problem's constraint. All these problems, except for the latter in integers, have polynomial time algorithms which may be viewed within a unifying framework of a proximity-scaling technique or a threshold technique. The complexity of many of these algorithms is furthermore best possible in that it matches lower bounds on the complexity of the respective problems. In general nonseparable optimization problems are shown to be considerably more difficult than separable problems. We compare the complexity of continuous versus discrete nonlinear problems and list some major open problems in the area of nonlinear optimization.

Achieving sustainable profit advantage, cost reduction and resource utilization are always a bottleneck for resource providers, especially when trying to meet the computing needs of resource hungry applications in mobile edge-cloud (MEC) continuum. Recent research uses metaheuristic techniques to allocate resources to large-scale applications in MECs. However, some challenges attributed to the metaheuristic techniques include entrapment at the local optima caused by premature convergence and imbalance between the local and global searches. These may affect resource allocation in MECs if continually implemented. To address these concerns and ensure efficient resource allocation in MECs, we propose a fruit fly-based simulated annealing optimization scheme (FSAOS) to serve as a potential solution. In the proposed scheme, the simulated annealing is incorporated to balance between the global and local search and to overcome its premature convergence. We also introduce a trade-off factor to allow application owners to select the best service quality that will minimize their execution cost. Implementation of the FSAOS is carried out on EdgeCloudSim Simulator tool. Simulation results show that the FSAOS can schedule resources effectively based on tasks requirement by returning minimum makespan and execution costs, and achieve better resource utilization compared to the conventional fruit fly optimization algorithm and particle swarm optimization. To further unveil how efficient the FSAOSs, a statistical analysis based on 95% confidential interval is carried out. Numerical results show that FSAOS outperforms the benchmark schemes by achieving higher confidence level. This is an indication that the proposed FSAOS can provide efficient resource allocation in MECs while meeting customers’ aspirations as well as that of the resource providers.

Industrial energy systems are increasingly required to reduce operating costs and CO2 emissions while integrating variable renewable energy sources. Managing these objectives under uncertainty requires advanced optimization strategies capable of delivering reliable and real-time decisions. To address these challenges, this study focuses on the short-term operational planning of an industrial energy supply system using the rolling horizon approach (RHA). The RHA offers an effective framework to handle uncertainties by repeatedly updating forecasts and re-optimizing over a moving time window, thereby enabling adaptive and responsive energy management. To solve the resulting nonlinear and constrained optimization problem at each RHA iteration, we propose a novel hybrid algorithm that combines Bayesian optimization (BO) with the Interior Point OPTimizer (IPOPT). While global deterministic and stochastic optimization methods are frequently used in practice, they often suffer from high computational costs and slow convergence, particularly when applied to large-scale, nonlinear problems with complex constraints. To overcome these limitations, we employ the BO–IPOPT, integrating the global search capabilities of BO with the efficient local convergence and constraint fulfillment of the IPOPT. Applied to a large-scale real-world case study of a food and cosmetic industry in Germany, the proposed BO–IPOPT method outperformed state-of-the-art solvers in both solution quality and robustness, achieving up to 97.25%-better objective function values at the same CPU time. Additionally, the influence of key parameters, such as forecast uncertainty, optimization horizon length, and computational effort per RHA iteration, was analyzed to assess their impact on system performance and decision quality.

This paper formulates resource allocation for decode-and-forward (DF) relay assisted multi-cell orthogonal frequency division multiple (OFDM) networks as an optimization problem taking into account of inter-cell interference and users fairness. To maximize the transmit rate of system we propose a joint interference coordination, subcarrier and power allocation algorithm. To reduce the complexity, this semi-distributed algorithm divides the primal optimization into three sub-optimization problems, which transforms the mixed binary nonlinear programming problem (BNLP) into standard convex optimization problems. The first layer optimization problem is used to get the optimal subcarrier distribution index. The second is to solve the problem that how to allocate power optimally in a certain subcarrier distribution order. Based on the concept of equivalent channel gain (ECG) we transform the max-min function into standard closed expression. Subsequently, with the aid of dual decomposition, water-filling theorem and iterative power allocation algorithm the optimal solution of the original problem can be got with acceptable complexity. The third sub-problem considers dynamic co-channel interference caused by adjacent cells and redistributes resources to achieve the goal of maximizing system throughput. Finally, simulation results are provided to corroborate the proposed algorithm.

A constructive heuristic algorithm (CHA) to solve distribution system planning (DSP) problem is presented. The DSP is a very complex mixed binary nonlinear programming problem. A CHA is aimed at obtaining an excellent quality solution for the DSP problem. However, a local improvement phase and a branching technique were implemented in the CHA to improve its solution. In each step of the CHA, a sensitivity index is used to add a circuit or a substation to the distribution system. This sensitivity index is obtained by solving the DSP problem considering the numbers of circuits and substations to be added as continuous variables (relaxed problem). The relaxed problem is a large and complex nonlinear programming and was solved through an efficient nonlinear optimization solver. Results of two tests systems and one real distribution system are presented in this paper in order to show the ability of the proposed algorithm.

Feature selection is an important part of data mining, image recognition and other fields. The efficiency and accuracy of classification algorithm can be improved by selecting the best feature subset. The classical feature selection technology has some limitations, and heuristic optimization algorithm for feature selection is an alternative method to solve these limitations and find the optimal solution. In this paper, we proposed a Multi Strategy Grey Wolf Optimizer algorithm (MSGWO) based on random guidance, local search and subgroup cooperation strategies for feature selection, which solves the problem that the traditional grey wolf optimizer algorithm (GWO) is easy to fall into local optimization with a single search strategy. Among them, the random guidance strategy can make full use of the random characteristics to enhance the global search ability of the population, and the local search strategy makes grey wolf individuals make full use of the search space around the current best solution, and the subgroup cooperation strategy is very important to balance the global search and local search of the algorithm in the iterative process. MSGWO algorithm cooperates with each other in three strategies to update the location of grey wolf individuals, and enhances the global and local search ability of grey wolf individuals. Experimental results show that MSGWO can quickly find the optimal feature combination and effectively improve the performance of the classification model.