Jonas Mockus

Application of Bayesian approach to numerical methods of global and stochastic optimization

In this paper a review of application of Bayesian approach to global and stochastic optimization of continuous multimodal functions is given. Advantages and disadvantages of Bayesian approach (average case analysis), comparing it with more usual minimax approach (worst case analysis) are discussed. New interactive version of software for global optimization is discussed. Practical multidimensional problems of global optimization are considered

Stochastic 16-State Model of Voltage Gating of Gap-Junction Channels Enclosing Fast and Slow Gates

Gap-junction (GJ) channels formed of connexin (Cx) proteins provide a direct pathway for electrical and metabolic cell-cell interaction. Each hemichannel in the GJ channel contains fast and slow gates that are sensitive to transjunctional voltage (Vj). We developed a stochastic 16-state model (S16SM) that details the operation of two fast and two slow gates in series to describe the gating properties of homotypic and heterotypic GJ channels. The operation of each gate depends on the fraction of Vj that falls across the gate (VG), which varies depending on the states of three other gates in series, as well as on parameters of the fast and slow gates characterizing 1), the steepness of each gates open probability on VG; 2), the voltage at which the open probability of each gate equals 0.5; 3), the gating polarity; and 4), the unitary conductances of the gates and their rectification depending on VG. S16SM allows for the simulation of junctional current dynamics and the dependence of steady-state junctional conductance (gj,ss) on Vj. We combined global coordinate optimization algorithms with S16SM to evaluate the gating parameters of fast and slow gates from experimentally measured gj,ss-Vj dependencies in cells expressing different Cx isoforms and forming homotypic and/or heterotypic GJ channels.

Bayesian heuristic approach to global optimization and examples

The traditional numerical analysis considers optimization algorithms which guarantee some accuracy for all functions to be optimized. This includes the exact algorithms. Limiting the maximal error requires a computational effort that in many cases increases exponentially with the size of the problem (Horst and Pardalos, 1995, Handbook of Global Optimization, Kluwer). That limits practical applications of the worst case analysis. An alternative is the average case analysis where the average error is made as small as possible (Calvin and Glynn, 1997, J. Appl. Prob., 32: 157). The average is taken over a set of functions to be optimized. The average case analysis is called the Bayesian Approach (BA) (Diaconis, 1988, Statistical Decision Theory and Related Topics, Springer; Mockus and Mockus, 1987, Theory of Optimal Decisions, Nauk, Lithuania). Application of BA to optimization of heuristics is called the Bayesian Heuristic Approach (BHA) (Mockus, 2000, A Set of Examples of Global and Discrete Optimization, Kluwer). In this paper a short presentation of the basic ideas of BHA (described in detail in Mockus (1989), Bayesian Approach to Global Optimization, Kluwer and Mockus (2000), A Set of Examples of Global and Discrete Optimization, Kluwer) is given using the knapsack problem as an example. The application potential is illustrated by the school scheduling example. In addition the new heuristic algorithm for solving a bimatrix game problem is investigated. The results ae applied while solving real life optimization problems and also as examples for distance graduate level studies of the theory of games and markets in the Internet environment.

On simulation of optimal strategies and Nash equilibrium in the financial market context

Unlike physical time series, stock market prices may be affected by the predictions made by market participants with conflicting interests. This is the domain of game theory. Therefore, we propose a Stock Exchange Game Model (SEGM) to model this phenomenon. In SEGM, player strategies are to set their buying and selling levels for the next iteration via the autoregressive model AR(p) of order p selected by minimizing deviations from Nash Equilibrium (NE). NE represents the assumption of optimal behavior by market participants. The objective of SEGM is to simulate financial and other time series that are affected by predictions of the participants and to test the assumption of optimal player behavior, using a ‘virtual’ stock exchange. The simulation of SEGM suggests that NE is close to the Wiener model. This is a new explanation of the Random Walk (RW) model of the efficient market theory. To compare the simulation results with real data, the efficient market hypothesis was also tested, using financial time series of eight assets. The SEGM software is implemented in Java applets and can be run using a browser with Java support. The main web site is in http://soften.ktu.lt/~mockus.

A System for Distance Studies and Applications of Metaheuristics

The efficiency of metaheuristics depends on parameters. Often this relation is defined by statistical simulation and have many local minima. Therefore, methods of stochastic global optimization are needed to optimize the parameters. The traditional numerical analysis considers optimization algorithms that guarantee some accuracy for all functions to be optimized. This includes the exact algorithms. Limiting the maximal error requires a computational effort that often increases exponentially with the size of the problem [Horst and Pardalos (1995), Handbook of Global Optimization, Kluwer Academic Publisher, Dordrecht/Boston/London]. That limits practical applications. An alternative is the average analysis where the expected error is made as small as possible [Calvin and Zilinskas (2000), JOTA Journal of Optimization Theory and Applications, 106, 297–307]. The average is taken over a set of functions to be optimized. The average analysis is called the Bayesian Approach (BA) [Diaconis (1988), Statistical Decision Theory and Related Topics, Springer-Verlag, Berlin, pp. 163–175, Mockus and Mockus (1987), Theory of Optimal Decision, Vol. 12, Institute of Mathematics and Cybernetics, Akademia Nauk Lithuanian SSR, Vilnius, Lithuania pp. 57–70]. Application of BA to optimization of heuristics is called the Bayesian Heuristic Approach (BHA) [Mockus (2000), A Set of Examples of Global and Discrete Optimization: Application of Bayesian Heuristic Approach, Kluwer Academic Publishers, Dordrecht, ISBN 0-7923-6359-0]. If the global minimum is known then the traditional stopping condition is applied: stop if the distance to the global minimum is within acceptable limits. If the global minimum is not known then the different approach is natural: minimize the average deviation during the fixed time limit because there is no reason to stop before. If the distance from the global minimum is not known the efficiency of method is tested by comparing with average results of some other method. “Pure” Monte Carlo is a good candidate for such comparison because it converges and does not depend on parameters that can be adjusted to a given problem by using some expert knowledge or additional test runs. In this paper a short presentation of the basic ideas of BHA [described in detail in Mockus (2000, A Set of Examples of Global and Discrete Optimization: Application of Bayesian Heuristic Approach, Kluwer Academic Publishers, Dordrecht, ISBN 0-7923-6359-0) and Mockus (1989, Bayesian Approach to Global Optimization, Kluwer Academic Publishers, Dordrec ht-London-Boston)] is given. The simplest knapsack problem is for initial explanation of BHA. The possibilities of application are illustrated by a school scheduling problem and other examples. Designed for distance graduate studies of the theory of games and markets in the Internet environment. All the algorithms are implemented as platform independent Java applets or servlets therefore readers can easily verify and apply the results for studies and for real life heuristic optimization problems. To address this idea, the paper is arranged in a way convenient for the direct reader participation. Therefore, a part of the paper is written as some “user guide”. The rest is a short description of optimization algorithms and models. All the remaining information is on web-sites, for example http://pilis.if.ktu.lt/~mockus.

Investigation of Data Regularization and Optimization of Timetables by Lithuanian High Schools Example

In practice, we must first assign teachers and students to subject-groups for school applications. In the Lithuanian high schools, the number of subject-groups can be very large, since students are free to select just a small subset of optional subjects.