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A particular discrete dynamical program used as a model for one specific situation in chess involving knights and rooks 22nd EUROPEAN CONFERENCE ON OPERATIONS.

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Presentation on theme: "A particular discrete dynamical program used as a model for one specific situation in chess involving knights and rooks 22nd EUROPEAN CONFERENCE ON OPERATIONS."— Presentation transcript:

1 A particular discrete dynamical program used as a model for one specific situation in chess involving knights and rooks 22nd EUROPEAN CONFERENCE ON OPERATIONS RESEARCH Prague, July 8-11, 2007 Camilo Ortiz, René Meziat Departamento de Matemáticas Universidad de los Andes Bogotá, Colombia 4/8/2019 Knight Problem

2 What is the Knight Problem intended for?
4/8/2019 Knight Problem

3 What is the Knight Problem intended for?
4/8/2019 Knight Problem

4 Essential features of this problem
It can be modeled as a dynamical program with discrete state variables and discrete control variables We exploit the symmetries of the knight movement to reduce the number of variables involved in the dynamical program To overcome the discrete nature of all variables, we propose a semidefinite relaxation based in algebraic and trigonometric moments 4/8/2019 Knight Problem

5 Parameters of the Knight Problem
Number of steps: N Dimension of the board side: n Allowed set of movements of the knight on the chess board: Mk={(1, 2), (2, 1), (−1, 2), (−2, 1), (1,−2), (2,−1), (−1,−2), (−2,−1)} Sets of x and y coordinates of the positions of m enemy rooks: PX = {p1x, , pmx}. PY = {p1y, , pmy}. 4/8/2019 Knight Problem

6 State and Control Variables with their respective constraints
(xt,yt) give the positions of the knight on the chess board at every time t=0,. . .,N. Constraints of the state variables: xt ∊ {1,. . .,n}\PX for t=0,. . .,N yt ∊ {1,. . .,n}\PY for t=0,. . .,N (ut,vt) represents a two dimensional control variable for every time t=0,. . .,N-1. In order to model the knight movements we need the following constraints on the control variables (ut,vt) ∊ Mk, for t=0,. . .,N-1. 4/8/2019 Knight Problem

7 The discrete dynamical program model for the Knight Problem
A dimensional reduction on the knight motion Quadratic Objective Dynamical System Discrete 1-dim State Constraints Discrete 2-dim Control Constraint 4/8/2019 Knight Problem

8 A dimensional reduction on the knight motion
4/8/2019 Knight Problem

9 A dimensional reduction on the knight motion
A trigonometric representation of the movements of the knight reduces the dimension of the control variables: 4/8/2019 Knight Problem

10 The discrete dynamical program model for the Knight Problem
Discrete 2-dim Constraint Discrete 1-dim Constraints Dynamical System Quadratic Objective Quadratic Objective Dynamical System Discrete 1-dim Constraints 4/8/2019 Knight Problem

11 Steps toward a convex exact relaxation of the problem based on algebraic and trigonometric moments
Discrete constraints can be transformed easily into polynomial constraints. One dimensional polynomial constraints can be expressed as linear matrix inequalities. Convex quadratic objective function can be expressed as a linear objective function. 4/8/2019 Knight Problem

12 1. transforming discrete constraints to polynomial constraints
state discrete constraints we define the algebraic polynomials we obtain equivalent polynomial constraints 4/8/2019 Knight Problem

13 1. transforming discrete constraints to polynomial constraints
control discrete one dimensional constraint we define the trigonometric polynomial we obtain the trigonometric polynomial constraint 4/8/2019 Knight Problem

14 1. transforming discrete constraints to polynomial constraints
Dynamical System Quadratic Objective Quadratic Objective Dynamical System Polynomial Constraints 4/8/2019 Knight Problem

15 Steps toward a convex exact relaxation of the problem based on algebraic and trigonometric moments
Discrete constraints can be transformed easily into polynomial constraints. One dimensional polynomial constraints can be expressed as linear matrix inequalities. Convex quadratic objective function can be expressed as a linear objective function. 4/8/2019 Knight Problem

16 Functional basis Linear Combinations
2. polynomial constraints to linear matrix inequalities Functional basis Algebraic Functional Basis for State Variables Trigonometric Functional Basis for Control Variables Linear Combinations 4/8/2019 Knight Problem

17 Algebraic moments Hankel Matrix
2. polynomial constraints to linear matrix inequalities Algebraic moments Hankel Matrix Positive Semidefinite 4/8/2019 Knight Problem

18 Trigonometric moments
2. polynomial constraints to linear matrix inequalities Trigonometric moments Toeplitz matrix Positive Semidefinite Hermitian symmetry 4/8/2019 Knight Problem

19 2. polynomial constraints to linear matrix inequalities
Linearization and convexification of the problem: Change of variables Plus positive semidefinite Hankel and Toeplitz matrices. 4/8/2019 Knight Problem

20 2. polynomial constraints to linear matrix inequalities
Objective Polynomial Constraints Dynamical System Quadratic Objective Dynamical System Convex Semidefinite Program Linear Constraints Hankel Semidefinite Constraints Toeplitz Semidefinite Constraints 4/8/2019 Knight Problem

21 Steps toward a convex exact relaxation of the problem based on algebraic and trigonometric moments
Discrete constraints can be transformed easily into polynomial constraints. One dimensional polynomial constraints can be expressed as linear matrix inequalities. Convex quadratic objective function can be expressed as a linear objective function plus a linear matrix inequality. 4/8/2019 Knight Problem

22 Computational solution
SeDuMi: from Advanced Optimization Lab of McMaster University. AMD Turion 64 processor 1.6 Ghz 896 Mb 4/8/2019 Knight Problem

23 Solving the Knight Problem
Numerical results for the solution depicted in orange. Numerical results for the solution depicted in blue. We find two solutions for the following instance with two rooks. n=8 N=6 2 rooks 4/8/2019 Knight Problem

24 Solving the Knight Problem
We find a unique solution for the following instance with three rooks. Numerical solution n=8 N=6 3 rooks 4/8/2019 Knight Problem

25 Conclusions This work is a very interesting application of the method of moments to dynamical programs with discrete variables. We open many possibilities to solve discrete optimal control problems in Operations Research. Our proposal is restricted to state of the art solvers for semidefinite programming. When the problem involves variables in several dimensions we must use a different approach by following an iterative scheme. 4/8/2019 Knight Problem

26 References Lasserre, J., Global optimization with polynomials and the problem of moments, SIAM J. Optim., vol 11, , 2001. Lasserre, J., Semidefinite programming vs lp relaxations for polynomial programming, J. of Math. of Operations Research, vol 27, , 2002. Henrion D. and J.B. Lasserre GloptiPoly : Global Optimization over Polynomials with Matlab and SeDuMi., ACM Trans. Math. Soft., 29, 165–194,2003. Lasserre, J. and Prieto-Rumeau T., SDP vs. LP Relaxations for the Moment Approach in Some Performance Evaluation Problems., Stoch. Models,20, 439–456, 2004. Lasserre, J., SOS approximations of polynomials nonnegative on a real algebraic set, SIAM J. Optim., 16, 610–628, 2005. Lasserre, J., A sum of squares approximation of nonnegative polynomials, SIAM, J. Optim., 16, 751–765, 2006. Nesterov, Y., Squared functional systems and optimization problems in High Performance Optimization, Frenk, H., K. Roos and T. Terlaky, eds., Kluwer, 2000. Curto, R. and L.A. Fialkow, Recursiveness, positivity and truncated moment problems, Houston Journal of Mathematics, Vol 17, p , 1991. 4/8/2019 Knight Problem

27 References Laurent, M., Revisiting two theorems of Curto and Fialkow on moment matrices, Proceedings of the AMS, 133, no. 10, 2965–2976, 2005. Laurent, M. and D. Jibetean, Semidefinite approximations for global unconstrained polynomial optimization, SIAM Journal on Optimization, 16(2), 490–514, 2005. Henrion, Didier; Garulli, Andrea (Eds.), Positive Polynomials in Control Series, Lecture Notes in Control and Information Sciences, Vol. 312, 2005. Parrilo, P. and B. Sturmfels, Minimizing polynomial functions, in Algorithmic and quantitative real algebraic geometry, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, AMS., Vol. 60, 83–99, 2001. Parrilo, P. and S. Lall, Semidefinite programming relaxations and algebraic optimization in Control, European Journal of Control, Vol. 9, No. 2-3, 2003. Parrilo, P.,Semidefinite programming relaxations for semialgebraic problems, Mathematical Programming Ser. B, Vol. 96, No.2, 293–320, 2003. Prajna, S., Parrilo P. and A. Rantzer, Nonlinear control synthesis by convex optimization, IEEE Transactions on Automatic Control, Vol. 49, No. 2, 2004. Mazzaro, M.C., P.A. Parrilo and R.S. S´anchez Pe˜na, Robust identification toolbox, Latin American Applied Research, Vol. 34, No. 2, 91–100, 2004. K. Gatermann and P.A. Parrilo, Symmetry groups, semidefinite programs,and sums of squares, Journal of Pure and Appl. Algebra, Vol. 192, No. 1-3, 95–128, 2004. 4/8/2019 Knight Problem

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