This paper presents a genetic algorithm (GA) for solving the traveling salesman problem (TSP). To construct a powerful GA, we use edge assembly crossover (EAX) and make substantial enhancements to it: (i) localization of EAX together with its efficient implementation and (ii) the use of a local search procedure in EAX to determine good combinations of building blocks of parent solutions for generating even better offspring solutions from very high-quality parent solutions. In addition, we develop (iii) an innovative selection model for maintaining population diversity at a negligible computational cost. Experimental results on well-studied TSP benchmarks demonstrate that the proposed GA outperforms state-of-the-art heuristic algorithms in finding very high-quality solutions on instances with up to 200,000 cities. In contrast to the state-of-the-art TSP heuristics, which are all based on the Lin–Kernighan (LK) algorithm, our GA achieves top performance without using an LK-based algorithm.

This paper presents a new evolutionary algorithm for function optimization named the distance-weighted exponential natural evolution strategies (DX-NES). DX-NES remedies two problems of a conventional method, the exponential natural evolution strategies (xNES), that shows good performance when it does not need to move the distribution for sampling individuals down the slope to the optimal point. The first problem of xNES is that the search efficiency deteriorates while the distribution moves down the slope of an ill-scaled function because it degenerates before reaching the optimal point. The second problem is that the settings of learning rates are inappropriate because they do not taking account of some factors affecting the estimate accuracy of the natural gradient. We compared the performance of DX-NES with that of xNES and CMA-ES on typical benchmark functions and confirmed that DX-NES outperformed the xNES on all the benchmark functions and that DX-NES showed better performance than CMA-ES on the almost all functions except the k-tablet function.

In our previous study, we have proposed Adaptive Weighted Aggregation (AWA), a framework of multi-starting optimization methods based on scalarization for solving multi objective function optimization problems. The experiments in the proposal show that AWA outperforms conventional multi starting descent methods at coverage of solutions. However, the suitable termination condition for AWA has not been understood. Coverage of AWA's solutions and computational cost of AWA strongly depends on the termination condition. In this paper, we derive the necessary and sufficient iteration count to achieve high coverage and the number of approximate solutions generated until AWA stops. Numerical experiments show that AWA still achieves better coverage than the conventional methods under the derived termination condition.

Adaptive Weighted Aggregation (AWA) is a frame work of multi-starting optimization methods based on scalarization for solving multiobjective function optimization problems. It progressively generates new solutions to refine the approximation of the Pareto set or the Pareto front by the subdivision, and iteratively estimates the appropriate weight vector for scalarization in each search by the weight adaptation. Our recent study shows that AWA's solution set combinatorially increases for the number of objectives. In this paper, we propose a new subdivision and weight adaptation scheme of AWA to improve its scalability. Numerical experiments show the effectiveness of the proposed method.

In this paper, we propose a new framework taking account of multi-funnel functions for Real-coded Genetic Algorithms (RCGAs). In the continuous function optimization, Evolutionary Algorithms (EAs) are one of the most effective optimization methods. However, most conventional EAs, such as RCGAs and CMA-ES, work efficiently on functions with big-valley landscape and they deteriorate on the multi-funnel functions. Innately Split Model (ISM) has been proposed as a framework of GAs for multi-funnel functions and outperforms conventional GAs on this kind of functions. However, ISM is considered to have two problems in terms of efficiency of the search and difficulty of parameter settings. Our framework repeats a search by RCGAs as ISM does and has two effective mechanisms to remedy the two problems of ISM. We conducted experiments on benchmark functions with multi-funnel and big valley landscapes and our framework outperformed conventional EAs, Multi-start RCGA (MS-RCGA), Multi-start CMA-ES (MS CMA-ES) and ISM, on the multi-funnel functions. Our frame work achieved as good performance as MS-RCGA and MS CMA-ES on the big-valley function where ISM significantly deteriorates.

The multi-starting descent method is a promising approach to unimodal multiobjective function optimization problems because of its precision of obtained solutions. Descent methods can be classified into two categories; the multiobjective descent method directly using the Jacobian matrix of objective functions and the scalarized descent method using the gradient of a scalarized objective function. In the multiobjective descent method and the scalarized descent method, a convergent point depends on an initial solution and a weight vector, respectively. However, it is difficult to choose appropriate initial solutions or weight vectors for obtaining widely and evenly distributed solutions. In order to remedy the problems of the conventional methods, we propose a multi-starting scalarized descent method named AWA that employs the Chebyshev norm method as a scalarization method and an adaptive scheme of weight vectors for the scalarization method. We show the effectiveness of the proposed method through some experiments.

Real-Coded Genetic Algorithms (RCGAs) have been extensively studied for last two decades because RCGAs have advantages over conventional continuous function optimization methods when multimodal functions are optimized. Innately Split Model (ISM) is one of promising approaches to enhance RCGAs where a set of population groups are evolved in parallel and groups are re-initialized if two groups searches a similar region (it is called redundant searches). In this paper, we propose a new strategy for the re-initialization of groups to improve the performance of ISM. In our method, redundant searches are detected by using the information of the search histories of the groups. This information is called traps and is stored as a set of hyper-ellipsoids representing the distributions of the previous groups. We demonstrate that the proposed method is robust and superior to the original ISM.

This paper investigates theoretically the convergence properties of the stochastic algorithms of a class including both CMAESs and EDAs on constrained minimization of continuously differentiable functions. We are interested in algorithms that do not get stuck on a slope of the function, but converge only to local optimal points. Convergence to a point that is neither a stationary point of the function nor a boundary point is evidence that the convergence properties are not well behaved. We investigate what properties are necessary/sufficient for the algorithm to avoid this type of behavior, i.e., what properties are necessary for the algorithm to converge only to local optimal points of the function. We also investigate the analogous conditions on the parameters of two variants of modern EC-based stochastic algorithms, namely, a CMAES employing rank-μ update and an EDA known as EMNAglobal. The comparison between the apparently similar two systems shows that they have significantly different theoretical behaviors. This result presents us with an insight into the way we design well-behaved optimization algorithms.

In this paper, we propose an efficient algorithm for estimating the natural policy gradient with parameter-based exploration; this algorithm samples directly in the parameter space. Unlike previous methods based on natural gradients, our algorithm calculates the natural policy gradient using the inverse of the exact Fisher information matrix. The computational cost of this algorithm is equal to that of conventional policy gradients whereas previous natural policy gradient methods have a prohibitive computational cost. Experimental results show that the proposed method outperforms several policy gradient methods.

The purpose of this paper is to propose a new real-coded genetic algorithm (RCGA) named Networked Genetic Algorithm (NGA) that intends to find multiple optima simultaneously in deceptive globally multimodal landscapes. Most current techniques such as niching for finding multiple optima take into account big valley landscapes or non-deceptive globally multimodal landscapes but not deceptive ones called UV-landscapes. Adaptive Neighboring Search (ANS) is a promising approach for finding multiple optima in UV-landscapes. ANS utilizes a restricted mating scheme with a crossover-like mutation in order to find optima in deceptive globally multimodal landscapes. However, ANS has a fundamental problem that it does not find all the optima simultaneously in many cases. NGA overcomes the problem by an adaptive parent-selection scheme and an improved crossover-like mutation. We show the effectiveness of NGA over ANS in terms of the number of detected optima in a single run on Fletcher and Powell functions as benchmark problems that are known to have UV-landscapes. We also analyze the behavior of NGA to confirm that the adaptive parent-selection scheme contributes the performance of NGA.

Premature convergence is one of the most notable obstacles that GAs face with. Once it happens, GAs cannot generate candidate solutions globally and the solutions are finally captured by local minima. To overcome it, we propose a mechanism that indirectly controls the variety of the population. It is realized by adapting the expansion rate parameter of crossovers, which determines the variance of the crossover distribution. The resulting algorithm is called adaptation of expansion rate (AER). The performance of the proposed methods is compared to an existing GA on several benchmark functions including functions whose landscape have ridge or multimodal structure. On these functions, existing GAs are likely to lead to premature convergence. The experimental result shows our approach outperforms the existing one on deceptive functions without disturbing the performance on comparatively easy problems.

Link analysis methods have been used successfully for knowledge discovery from the link structure of mutually linking entities. Existing link analysis methods have been inherently designed based on the fact that the entire link structure of the target graph is observable such as public web documents; however, link information in graphs in the real world, such as human relationship or economic activities, is rarely open to public. If link analysis can be performed using graphs with private links in a privacy-preserving way, it enables us to rank entities connected with private ties, such as people, organizations, or business transactions. In this paper, we present a secure link analysis for graphs with private links by means of cryptographic protocols. Our solutions are designed as privacy-preserving expansions of well-known link analysis methods, PageRank and HITS. The outcomes of our protocols are completely equivalent to those of PageRank and HITS. Furthermore, our protocols theoretically guarantee that the private link information possessed by each node is not revealed to other nodes. %We demonstrate the efficiency of our solution by experimental studies, comparing with existing solutions, such as secure function evaluation, decentralized spectral analysis, and privacy-preserving link-analysis.

This paper aims the design of efficient and effective optimization algorithms for function optimization. This paper presents a new framework of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). Recent studies modified the CMA-ES from the viewpoint of covariance matrix adaptation and resulted in drastic reduction of the number of generations. In addition to their modification, this paper modifies the CMA-ES from the viewpoint of step size adaptation. The main idea of modification is semantically specializing functions of covariance matrix adaptation and step size adaptation. This new method is evaluated on 8 classical unimodal and multimodal test functions and the performance is compared with standard CMA-ES. The experimental result demonstrates an improvement of the search performances in particular with large populations. This result is mainly because the proposed Hybrid-SSA instead of the existing CSA can adjust the global step length more appropriately under large populations and function specialization helps appropriate adaptation of the overall variance of the mutation distribution.

We consider the problem of distributed reinforcement learning (DRL) from private perceptions. In our setting, agents' perceptions, such as states, rewards, and actions, are not only distributed but also should be kept private. Conventional DRL algorithms can handle multiple agents, but do not necessarily guarantee privacy preservation and may not guarantee optimality. In this work, we design cryptographic solutions that achieve optimal policies without requiring the agents to share their private information.

Although multi-objective GA (MOGA) is an efficient multi-objective optimization (MOO) method, it has some limitations that need to be tackled, which include unguaranteed uniformity of solutions and uncertain finding of periphery of Pareto-optimal solutions. It has been shown that, on bi-objective problems, which are the subject of this paper, local Pareto-optimal solutions form curves. In this case, some of the limitations of MOGA can be resolved by sampling the curves uniformly in the variable space and in the objective space. This paper proposes Pareto Path Following (PPF) which does the sampling by extending the framework of Numerical Path Following, verifies that PPF exhibits the desired behaviors, and addresses the extension of PPF for problems with more than two objective functions.Application of PPF is not limited to refinement of solutions obtained with MOGA. PPF makes it natural to have a local Pareto-optimal solution curve as the unit of search, which leads to curve-based MOGA. PPF also enables examination of which Pareto-optimal solution curves are found by MOO methods, and performance metrics based on it can be defined. This paper proposes these applications of PPF in MOGA and compares standard MOGA and curve-based MOGA using the metrics to reveal their characteristics.

We propose a protocol for a local search and a genetic algorithm for the distributed traveling salesman problem (TSP). In the distributed TSP, information regarding the cost function such as traveling costs between cities and cities to be visited are separately possessed by distributed parties and both are kept private each other. We propose a protocol that securely solves the distributed TSP by means of a combination of genetic algorithms and a cryptographic technique, called the secure multiparty computation. The computation time required for the privacy preserving optimization is practical at some level even when the city-size is more than a thousand.

This paper describes a GA based lazy approach toward reinforcement learning. This approach employs data-driven policy, which is composed of an instance set and an instance-based action selector. This feature provides a number of advantages. However some difficulties remain uninvestigated. One of them is the huge and complicated search space. We have an idea that preserving characteristics of the GA population and introducing new characteristics can overcome these difficulties. On the basis of this idea, we propose two genetic operators; Binomial Distribution Crossover (BDX) and iterated refreshment. The BDX generates the descendants inheriting the parents’ characteristics and the iterated refreshment introduces new characteristics greedily. The GA powered by these operators was applied to the benchmark tasks to demonstrate the ability. Each operator also was investigated and discussed from the various perspectives. Finally, we provide the preferable parameter settings for our method.

Hybridization with local search (LS) is known to enhance the performance of genetic algorithms (GA) in single objective optimization and have also been studied in the multiobjective combinatorial optimization literature. In most such studies, LS is applied to the solutions of each generation of GA, which is the scheme called "GA with LS" herein. Another scheme, in which LS is applied to the solutions obtained with GA, has also been studied, which is called "GA then LS" herein. It seems there is no consensus in the literature as to which scheme is better, let alone the reasoning for it. The situation in the multiobjective function optimization literature is even more unclear since the number of such studies in the field has been small.This paper, assuming that objective functions are differentiable, reveals the reasons why GA is not suitable for obtaining solutions of high precision, thereby justifying hybridization of GA and LS. It also suggests that the hybridization scheme which maximally exploits both GA and LS is GA then LS. Experiments conducted on many benchmark problems verified our claims.

Genetic Algorithm (GA) is known as a potent multiobjective optimization method, and the effectiveness of hybridizing it with local search (LS) has recently been reported in the literature. However, there is a relatively small number of studies on LS methods for multiobjective function optimization. Although each of the existing LS methods has some strong points, they have respective drawbacks such as high computational cost and inefficiency in improving objective functions. Hence, a more effective and efficient LS method is being sought, which can be used to enhance the performance of the hybridization.Defining Pareto descent directions as descent directions to which no other descent directions are superior in improving all objective functions, this paper proposes a new LS method, Pareto Descent Method (PDM), which finds Pareto descent directions and moves solutions in such directions thereby improving all objective functions simultaneously. In the case part or all of them are infeasible, it finds feasible Pareto descent directions or descent directions as appropriate. PDM finds these directions by solving linear programming problems, which is computationally inexpensive. Experiments have shown PDM's superiority over existing methods.

The function optimization is one of the most important optimization problems. In approaches to function optimization by evolutionary computation, a real-coded genetic algorithm, UNDX+MGG, shows good performance on multimodal functions with epistasis among parameters. However, UNDX+MGG has a problem that its performance is good on functions with big valley structures but deteriorates on those with the UV structures. On the other hand, ISM shows good performance on functions with the UV structures. However, ISM has two problems that 1) it fails in search when the region of the V valley including the optimum is very narrow and 2) its performance deteriorates on functions with big valley structures. In this paper, we propose a new evolutionary algorithm that aims at overcoming the problems of UNDX+MGG and ISM and examine its effectiveness through some experiments.

We proposed a selection scheme called sample-based crowding, which is aimed to improve the performance of genetic algorithms for multimodal optimization in ill-scaled and locally multimodal domains. These domains can be problematic for conventional approaches, but are commonly found in real-world optimization problems. The principle of crowding is to apply a tournament selection to a parent-child pair with a high similarity. In the sample-based crowding, we determine such pairs based on a statistical comparison of the fitness values, which are sampled from the region between the pairs. Further, we take into account the ranks of the parents among the sampled values in the selection process, to determine their indispensability. These measurements are scale-invariant, which enables the proposed method to search a domain without presuming the distance between the optima or the scaling and the correlation of the variables. The proposed approach is evaluated in two benchmark problems with an ill-scaled and a locally multimodal landscape. The proposed method has a substantial advantage in terms of comprehensiveness compared to the conventional approaches, despite the additional cost of evaluations.

We propose a selection scheme called Fitness-based Neighbor Selection (FNS) for multimodal optimization. The FNS is aimed for ill-scaled and locally multimodal domain, both found in real-world numerical optimization problem.In FNS, selection is applied to parent-child pair that most likely belong to the same attractor. We determine such pair with statistical comparison of the fitness values sampled from region between the pairs, instead of conventional Euclidean distance. In addition, the ranks of a parent among sampled values are used to determine if the parent is replaceable. These measurements makes the algorithm scale-invariant thus robust in ill-scaled domain.

In this paper, we propose a GA model called Adaptive Isolation Model(AIM), for multimodal optimization. It uses a data clustering algorithm to detect clusters in GA population, which identifies the attractors in the fitness landscape. Then, subpopulations which makes-up the clusters are isolated and optimized independently. Meanwhile, the region of the isolated subpopulations in the original landscape are suppressed. The isolation increases comprehensiveness, i.e., the probability of finding weaker attractors, and the overall efficiency of multimodal search. The advantage of the AIM is that it does not require distance between the optima as a presumed parameter, as it is estimated from the variance/covariance matrix of the subpopulation.Further, AIM's behavior and efficiency is equivalent to basic GA in unimodal landscape, in terms of number of evaluation. Therefore, it is applied recursively to all subpopulations until they converge to a suboptima. This makes AIM suitable for locally-multimodal landscapes, which have closely located attractors that are difficult to distinguish in the initial run.The performance of AIM is evaluated in several benchmark problems and compared to iterated hill-climbing methods.

This paper presents a kernel density estimation method by means of real-coded crossovers. Estimation of density algorithms (EDAs) are evolutionary optimization techniques, which determine the sampling strategy by means of a parametric probabilistic density function estimated from the population. Real-coded Genetic Algorithm (RCGA) does not explicitly estimate any probabilistic distribution, however, the probabilistic model of the population is implicitly estimated by crossovers and the sampling strategy is determined by this implicit probabilistic model. Based on this understanding, we propose a novel density estimation algorithm by using crossovers as nonparametric kernels and apply this kernel density estimation to the Gaussian Mixture modeling. We show that the proposed method is superior in the robustness of the computation and in the accuracy of the estimation by the comparison of conventional EM estimation.

This paper presents the Real-coded Genetic Algorithms for high-dimensional ill-scaled structures, what is called, the k-tablet structure. The k-tablet structure is the landscape that the scale of the fitness function is different between a k-dimensional subspace and the orthogonal (n−k)-dimensional subspace. The search speed of traditional GAs degrades when a high dimensional k-tablet structure is included in the landscape of the fitness function. In this structure, offspring generated by crossovers are likely to spread wider region than the region where the parental population covers and this causes the stagnation of the search. To resolve this problem, we propose a new crossover LUNDX-m using only m-dimensional latent variables. The effectiveness of the proposal method is tested with several benchmark functions including k-tablet structures and we show that our proposed method performs better than traditional crossovers especially when the dimensionality n is higher than 100. As an example of a k-tablet structure in real world applications, we show that the lens design problem has a kind of k-tablet structures and that our proposed method also performs better than conventional crossovers in this problem.

When we use genetic algorithms to solve any type of problems, it is important to maintain the diversity of populations for avoiding early stage stagnation or falling into local minima. We propose an angular distance dependent alternation (ADDA) model as a generation alternation model on real-coded genetic algorithms (GA) to improve its performance by maintaining adequate diversity of populations. The basic concept of the ADDA is that all of offspring generated by crossover operations will be clustered by a corresponding parent based on the angular distance metric and will be transposed from the parent. We compare performance of the proposed alternation model with previous family based minimal generation gap (MGG) model and distance dependent alternation (DDA) model. Using with the multi-parental unimodal normal distribution crossover (UNDX-m), the ADDA model shows good performance on three typical benchmark problems.

Most real world problems contain complex and various constraints, and this goes for the sewerage system control problem, our target. For handling them, the penalty depending on the degree of violation is often used. However, additive penalty method (APM) often leads the fatal compromise to a local optimum, in this paper we introduce independent constraint satisfaction (ICS). Another difficulty of this problem is the inaccuracy of inflow forecasting, we show the eager re-scheduling is superior to the lazy one.

The design of control systems for large-scale and complex industrial plants involves numerous trade-off problems, such as costs, quality, environmental impact, safety, reliability, accuracy, and robustness. Some of these parameters are even conflicting. Thus, the use of a multidiscipline approach is suggested to satisfy these requirements in an acceptable and well-balanced manner, and a fusion of soft computing and hard computing appears to be a natural and practical choice. Although the state-of-the-art soft computing technology has distinguished features, the use of soft computing technology would be ineffective, if it is improperly fused with conventional hard computing technology and control processes. Proper fusion is key to success, and a general model of fusion is worth examining. In this paper, through a survey of published literature, a general fusion model and fusion topologies are shown at the system level as well as at the algorithm level.

Power plant start-up scheduling is aimed at minimizing the start-up time while limiting maximum turbine rotor stresses. This scheduling problem is highly nonlinear and has a number of local optima. In our previous research, we proposed an efficient search model: genetic algorithms (GAs) with enforcement operation to focus the search along the edge of the feasible space where the optimal schedule is supposed to stay. Based on a nonlinear dynamic simulation and a linear inverse calculation with the iteration method, the enforcement operation is applied to move schedules generated by GA toward the edge. We prove that the optimal schedule lies on the edge, ensuring that searching along the edge instead of the entire space can improve the search efficiency significantly without missing the optimum. Furthermore, we provide a theoretical setting equation for the inverse enforcement gains of the linear inverse calculation, intended to move schedules closer to the edge at each iteration of the enforcement operation. The theoretical setting equation is verified and discussed with the test results. We propose the theoretical setting equation with the test results as a guideline for the use of our proposed search model: GA with enforcement operation.

Many multi-objective evolutionary algorithms (MOEAs) have been proposed over the years. The main part of the most successful algorithms such as PESA, or NSGA-II, are the Pareto based selection strategy that decide survivors using dominance among individuals. However, does the Pareto based selection strategy always succeed in finding the Pareto optimal solutions? This paper shows a very simple example that can cause serious trouble for the Pareto based MOEAs. In such an instance, various solutions, which are apart from the true Pareto-optimums, are left as hardly-dominated solutions. We define such solutions as dominance resistant solutions (DRSs), and show a class of problems which produces DRSs easily. To cope with this difficulty we propose the /spl alpha/-domination strategy that relaxes the domination introducing a weak trade-off among objectives. With the /spl alpha/-domination strategy, the DRSs are effectively purged from the population.

Proposes a new real-coded genetic algorithm (GA) using the combination of two crossovers: UNDX-m (unimodal normal distribution crossover - modified) and EDX (extrapolation-directed crossover). The search region of UNDX-m tends to be biased toward the inside of the area that the population of the GA covers. Because of this search bias, the GA using UNDX-m causes stagnation of its search if the cost surface has a certain kind of structure - viz. the so-called ridge structure or multiple-peak structure. In order to compensate for this fault of UNDX-m, we propose a new crossover - EDX - which has an extrapolative search area, and we show its effectiveness through numerical experiments.

We propose a new population-based evolutionary algorithm which uses a real-coded representation and normal-distribution crossover-like mutation for generating the next searching points. This Gaussian distribution is formed based on the positional relationships between an individual and its neighbors, and is not carried with the self-adapting parameters as an inheritable trait. This algorithm causes the emergence of clusters of individuals within the population, as a result of the evolution of each individual, which does not have any actual intent to cluster. By searching independently, the emergent clusters introduce various solutions that include optima at the same time, even if the problem has strong local minima. The proposed method robustly solves a highly multi-modal 30-dimensional Fletcher-Powell function with a small population size.

Reinforcement learning is a kind of machine learning. It aims to adapt an agent to a given environment with a clue to a reward. In general, the purpose of a reinforcement learning system is to acquire an optimum policy that can maximize expected reward per action. However, it is not always important for any environment. Especially, if we apply reinforcement learning to engineering, we expect the agent to avoid all penalties. In Markov decision processes, we call a rule penalty if and only if it has a penalty or it can transit to a penalty state where it does not contribute to get any reward. After suppressing all penalty rules, we aim to make a rational policy whose expected reward per action is larger than zero. We propose the penalty avoiding rational policy making algorithm that can suppress any penalty as stable as possible and get a reward constantly. By applying the algorithm to the tick-tack-toe its effectiveness is shown.

The unimodal normal distribution crossover (UNDX) for the real-coded genetic algorithms (RCGA) proposed by Ono et al. (1997, 1998) shows an excellent performance in optimization problems of multi-modal and highly epistatic fitness functions in continuous search space. Further, theoretical analysis of the UNDX shows that the UNDX is a crossover operator that preserves the statistics such as the mean vector and the covariance matrix of the population well. The present paper proposes some design guidelines for crossover operators for RCGA. Then, based on these guidelines, a multi-parental extension of the UNDX is proposed so as to enhance its exploration ability. Performance of the extended UNDX is evaluated by numerical experiments.

In recent years, a reinforcement learning approach to build an agent's knowledge in a multi-agent world has prevailed when the reinforcement learning is applied to such a world, "a concurrent learning among the agents", "a perceptual aliasing", and "a designing rewards" are the most important problems to be considered. We have already confirmed that profit-sharing algorithm shows its robustness against these three problems through some experiments. In this paper, we focus on an advantage of profit-sharing compared to Q-learning through the simulations of controlling cranes where there exist the conflicts among the agents. The conflict resolution problem must become a bottle-neck in the multi-agent world if we approach to it by the top-down method. Similarly, Q-learning is also weak in this problem without exhaustive design of the rewards or detailed information about other agents. We present that profit-sharing method can be available to resolve it, through the results of some experiments on the controlling cranes problem.

Power plant start-up scheduling is aimed at minimizing the start-up time while limiting maximum turbine-rotor stresses. A shorter start-up time not only reduces fuel and electricity consumption during the start-up process, but also increases its capability of adapting to changes in electricity demand. The start-up scheduling problem can be formulated as a function optimization problem with constraints. We have constructed an efficient and robust search model-a genetic algorithm (GA) with an enforcement operation-which forces the search along the edge of the feasible space, where the optimal schedule is supposed to exist. However, this model has to perform a prior Monte Carlo test to obtain the enforcement gains used for the implementation of the enforcement operation. In this paper, we attempt to eliminate the Monte Carlo test by proposing a self-reliant search model by introducing a GA with an adaptive enforcement operation that can generate and adapt enforcement gains during the search process. The test results of this proposed model show that the overall number of time-consuming dynamic simulations for the constraints calculation can be reduced further, thus increasing the overall efficiency of finding the optimal or near-optimal schedules.