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Register a new userMarkov Chain Analysis of Evolution Strategies on a Linear Constraint Optimization Problem
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Added by
Alexandre Chotard
Publication date
2014
fields
Informatics Engineering
and research's language is
English
Authors
Alexandre Chotard
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This paper analyses a $(1,lambda)$-Evolution Strategy, a randomised comparison-based adaptive search algorithm, on a simple constraint optimisation problem. The algorithm uses resampling to handle the constraint and optimizes a linear function with a linear constraint. Two cases are investigated: first the case where the step-size is constant, and second the case where the step-size is adapted using path length control. We exhibit for each case a Markov chain whose stability analysis would allow us to deduce the divergence of the algorithm depending on its internal parameters. We show divergence at a constant rate when the step-size is constant. We sketch that with step-size adaptation geometric divergence takes place. Our results complement previous studies where stability was assumed.
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This paper addresses the development of a covariance matrix self-adaptation evolution strategy (CMSA-ES) for solving optimization problems with linear constraints. The proposed algorithm is referred to as Linear Constraint CMSA-ES (lcCMSA-ES). It uses a specially built mutation operator together with repair by projection to satisfy the constraints. The lcCMSA-ES evolves itself on a linear manifold defined by the constraints. The objective function is only evaluated at feasible search points (interior point method). This is a property often required in application domains such as simulation optimization and finite element methods. The algorithm is tested on a variety of different test problems revealing considerable results.
This paper considers nonlinear regular-singular stochastic optimal control of large insurance company. The company controls the reinsurance rate and dividend payout process to maximize the expected present value of the dividend pay-outs until the time of bankruptcy. However, if the optimal dividend barrier is too low to be acceptable, it will make the company result in bankruptcy soon. Moreover, although risk and return should be highly correlated, over-risking is not a good recipe for high return, the supervisors of the company have to impose their preferred risk level and additional charge on firm seeking services beyond or lower than the preferred risk level. These indeed are nonlinear regular-singular stochastic optimal problems under ruin probability constraints. This paper aims at solving this kind of the optimal problems, that is, deriving the optimal retention ratio,dividend payout level, optimal return function and optimal control strategy of the insurance company. As a by-product, the paper also sets a risk-based capital standard to ensure the capital requirement of can cover the total given risk, and the effect of the risk level on optimal retention ratio, dividend payout level and optimal control strategy are also presented.
Physarum polycephalum inspired algorithm (PPA), also known as the Physarum Solver, has attracted great attention. By modelling real-world problems into a graph with network flow and adopting proper equations to calculate the distance between the nodes in the graph, PPA could be used to solve system optimization problems or user equilibrium problems. However, some problems such as the maximum flow (MF) problem, minimum-cost-maximum-flow (MCMF) problem, and link-capacitated traffic assignment problem (CTAP), require the flow flowing through links to follow capacity constraints. Motivated by the lack of related PPA-based research, a novel framework, the capacitated physarum polycephalum inspired algorithm (CPPA), is proposed to allow capacity constraints toward link flow in the PPA. To prove the validity of the CPPA, we developed three applications of the CPPA, i.e., the CPPA for the MF problem (CPPA-MF), the CPPA for the MCFC problem, and the CPPA for the link-capacitated traffic assignment problem (CPPA-CTAP). In the experiments, all the applications of the CPPA solve the problems successfully. Some of them demonstrate efficiency compared to the baseline algorithms. The experimental results prove the validation of using the CPPA framework to control link flow in the PPA is valid. The CPPA is also very robust and easy to implement since it could be successfully applied in three different scenarios. The proposed method shows that: having the ability to control the maximum among flow flowing through links in the PPA, the CPPA could tackle more complex real-world problems in the future.
The development, assessment, and comparison of randomized search algorithms heavily rely on benchmarking. Regarding the domain of constrained optimization, the number of currently available benchmark environments bears no relation to the number of distinct problem features. The present paper advances a proposal of a scalable linear constrained optimization problem that is suitable for benchmarking Evolutionary Algorithms. By comparing two recent EA variants, the linear benchmarking environment is demonstrated.
Since the debut of Evolution Strategies (ES) as a tool for Reinforcement Learning by Salimans et al. 2017, there has been interest in determining the exact relationship between the Evolution Strategies gradient and the gradient of a similar class of algorithms, Finite Differences (FD).(Zhang et al. 2017, Lehman et al. 2018) Several investigations into the subject have been performed, investigating the formal motivational differences(Lehman et al. 2018) between ES and FD, as well as the differences in a standard benchmark problem in Machine Learning, the MNIST classification problem(Zhang et al. 2017). This paper proves that while the gradients are different, they converge as the dimension of the vector under optimization increases.
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