Global optimization using interval analysis

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As a result, to determine our optimal path we would want to use simulation - optimization to first understand the range of potential times it could take to go from one point to another represented by a probability distribution in this case rather than a specific distance and then optimize our travel decisions to identify the best path to follow taking that uncertainty into account.

Stochastic tunneling STUN is an approach to global optimization based on the Monte Carlo method - sampling of the function to be objectively minimized in which the function is nonlinearly transformed to allow for easier tunneling among regions containing function minima. Easier tunneling allows for faster exploration of sample space and faster convergence to a good solution.

Parallel tempering , also known as replica exchange MCMC sampling , is a simulation method aimed at improving the dynamic properties of Monte Carlo method simulations of physical systems, and of Markov chain Monte Carlo MCMC sampling methods more generally. The replica exchange method was originally devised by Swendsen, [2] then extended by Geyer [3] and later developed, among others, by Giorgio Parisi. Essentially, one runs N copies of the system, randomly initialized, at different temperatures.

Then, based on the Metropolis criterion one exchanges configurations at different temperatures. The idea of this method is to make configurations at high temperatures available to the simulations at low temperatures and vice versa. This results in a very robust ensemble which is able to sample both low and high energy configurations.

In this way, thermodynamical properties such as the specific heat, which is in general not well computed in the canonical ensemble, can be computed with great precision. Other approaches include heuristic strategies to search the search space in a more or less intelligent way, including:. For general considerations on the dimensionality of the domain of definition of the objective function:.

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This article includes a list of references , but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. December Learn how and when to remove this template message. Main article: Deterministic global optimization.

Main article: Cutting plane. Main article: Branch and bound. Main articles: Interval arithmetic and Set inversion. Main article: Real algebraic geometry.

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  7. Main article: Stochastic optimization. Main article: Monte Carlo method. Main article: Stochastic tunneling.

    Global Optimization Using Interval Analysis (Pure & Applied Mathematics)

    Main article: Parallel tempering. Deem Bibcode : JChPh. Earl and Michael W. Deem "Parallel tempering: Theory, applications, and new perspectives" , Phys. Sugita and Y. Okamoto Chemical Physics Letters. Bibcode : CPL Vision Through Optimization. Visual Reconstruction. MIT Press. Fisher III. Bayesian approach to global optimization: theory and applications. Kluwer Academic. Deterministic global optimization: R. Horst, H. Horst, P. Pardalos and N. Kluwer Academic Publishers, Iserles, ed. Mongeau, H. Karsenty, V.

    Hiriart-Urruty, Comparison of public-domain software for black box global optimization. Kluwer Academic Publishers, Dordrecht, This book also discusses stochastic global optimization methods. Jaulin, M. Kieffer, O. Didrit, E. Walter Applied Interval Analysis. Berlin: Springer.

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    Strongin, Ya. Sergeyev ,, Global optimization with non-convex constraints: Sequential and parallel algorithms , Kluwer Academic Publishers, Dordrecht. Sergeyev, R. Strongin, D. Lera Introduction to global optimization exploiting space-filling curves , Springer, NY. DOI: Bonvin, C.

    Global optimization using interval analysis.

    Georgakis, C. Pantelides, M. Barolo, M. Grover, D.

    Global optimization using interval analysis — the multi-dimensional case

    Rodrigues, R. Schneider, and D. Linking Models and Experiments. Vivek Dua and Pinky Dua. Yao Zhao and Mark A. Spencer D. Schaber, Joseph K. Scott, Paul I. Convergence-order analysis for differential-inequalities-based bounds and relaxations of the solutions of ODEs. Quadratic interpolation based teaching-learning-based optimization for chemical dynamic system optimization. Knowledge-Based Systems , , Gisela C. Ramadas, Edite M. Ramadas, Ana Maria A. Rocha, M. Fernanda P. Journal of Optimization , , David L. Global optimisation for dynamic systems using interval analysis.

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    7. Jai Rajyaguru, Mario E. Chebyshev model arithmetic for factorable functions. Journal of Global Optimization , 68 2 , Ana Maria A. Rocha, Marisa C. Martins, M. Costa, Edite M. Ketan Dinkar Sarode, V. Ravi Kumar, B. Embedded multiple shooting methodology in a genetic algorithm framework for parameter estimation and state identification of complex systems. Chemical Engineering Science , , Carlos Perez-Galvan, I.

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      David, L. Shiyuan Sun, Jianwei Li. Parameter estimation of methanol transformation into olefins through improved particle swarm optimization with attenuation function. Chemical Engineering Research and Design , 92 , Joseph K. Improved relaxations for the parametric solutions of ODEs using differential inequalities. Journal of Global Optimization , 57 1 , Journal of Optimization Theory and Applications , 3 , Kan Dai, Ning Wang.

      A hybrid DNA based genetic algorithm for parameter estimation of dynamic systems. Chemical Engineering Research and Design , 90 12 ,