LP Infeasibility - The Pursuit of Intelligent Discretization

Transcription

1 LP - The Pursuit of Intelligent Discretization WORKSHOP ON CONVEX AND REAL-TIME OPTIMIZATION August 19, 2016 tol@es.aau.dk Department of Electronic Systems

2 Agenda the the

3 Lyapunov Fcn for Stability Analysis 2 Classical Lyapunov criteria: V (x) >0 V (0) = 0, V (x) 0. Thus V is positive definite (PD) and V is either PS or positive semi-definite (PSD). the

4 Lyapunov Fcn for Stability Analysis Classical Lyapunov criteria: Current standard is to relax as for global analysis, or as V (x) >0 V (0) = 0, V (x) 0. V (x) Σ 2 V (x) Σ 2 3 the V (x) Σ i s i p i Σ 2 V (x) Σ i s i p i Σ 2 for local analysis using Putinar s Positivestellensatz.

5 Another Way 4 But PSD SOS. And solving the SOS programs require semi-definite programming (SDP). the We do something else. Our method relies on the Bernstein basis. It always results in local analysis, but is also always results in linear programming (LP).

6 Bernstein basis Polynomials Defined on a simplex σ: 1D is a line, 2D is a triangle, 3D and up is called a simplex D of degree 2 B 2 (2;0) B 2 (0;2) 5 the B 2 (1;1) x 1

7 Bernstein basis Polynomials D of degree 3 B 3 (3;0) B 3 (0;3) B 3 (2;1) B 3 (1;2) 6 the x 1

8 Bernstein basis Polynomials D of degree 4 B 4 (4;0) B 4 (0;4) B(3;1) 4 B 4 B(2;2) 4 (1;3) 7 the x 1

9 Bernstein basis Polynomials 8 the

10 Bernstein basis Polynomials 9 the

11 Bernstein basis Polynomials 10 the

12 Bernstein Coefficients Any polynomial can be described in the Bernstein basis by a linear combination of the Bernstein basis polynomials, Bα. d p = b α (p, d, σ)bα d α =d The Bernstein coefficients contain valuable information about the polynomial p. The basis polynomials are non-negative on the simplex. Thus, if the coefficients are positive then so is p on the simplex. 11 the

13 p Convex Hull & End-point Value Property b (3;1) b (1;3) the 5 b (4;0) p b (0;4) 2.5 b (2;2) x

14 V Classical Lyapunov criteria the x

15 Now to the Fun of it! V = CV αv Bα dv V = CV T B d V α V =d V V = CL ˆα Bˆḓ α = CL T Bˆd ˆα =ˆd CV 0 CV α V = 0 CL 0 CL ˆα = 0 14 the

16 Now to the Fun of it! V = CV αv Bα dv V = CV T B d V α V =d V V = CL ˆα Bˆḓ α = CL T Bˆd ˆα =ˆd 15 the CV 0 CV α V = 0 CL 0 CL ˆα = 0 CL =A CV

17 Linear Feasibility Problem min CV 0 s.t. l c A CV 0 0 CV u x Any CV solving the LP problem is a Lyapunov function for the vector field and certifies the local stability. does not suggest the converse! 16 the

18 The Vector Field ẋ 1 = x 3 1 x x 3 1 x 2 x x 2 1 x 2 2 8x 2 1 x 2 + 4x 2 1 x 1 x x 1 x 3 2 4x x 2 2 ẋ 2 = 9x 2 1 x x x 1 x 3 2 8x 1x 2 2 4x 1 x x 2 2 4x 2 Two variables, degree 5, origin is stable and it is the only equilibrium. Investigated on the box B = [±1] 2 with a Lyapunov function of degree d V = 2. the 17

19 The Solution the 18

20 the But if the box is expanded to B = [±1.75] 2 the resulting LP is unsolvable. Then what? 19

21 Bernstein s Theorem Bernstein s Theorem: If a polynomial p of degree d is positive on a simplex σ, then there exists a sub-division of σ into a collection K = {σ 1,, σ m } of finitely many simplices such that b(p, d, σ i ) > 0 i {1,, m}. the 20 This motivates the use of sub-division in order to obtain a solvable LP.

22 The Solution Sub-dividing into 8 simplices renders the LP solvable. the 21 But the number of design variables and constraints grow rapidly with increase in the number of simplices.

23 Adding Slack Variables By adding one non-negative slack variable on each simplex, and minimizing their sum, the problematic parts of the domain can be identified min CV,s i m i s i s.t. l c A CV Bs 0 0 CV u x 0 s i the 22 This singles out some simplices for further sub-division, and leaves the rest as they are.

24 The Solution And it works. the 23 Saves on the number of design variables and constraints, yet obtains a solvable LP.

25 Advertisement Everything so far is covered in the paper: the T. Leth, C. Sloth, and R. Wisniewski, "Lyapunov Function Synthesis - Algorithm and Software," MSC, Buenos Aries, Argentina,

26 New Stuff Since writing the paper, has gained my attention as the constraint identifying mechanism. (simplified): Given A and b defining an LP, exactly one of the two proposition it true: 1. x : Ax = b, x 0, 2. y : A T y 0, b T y > 0. If such a y exists it is a certificate of infeasibility of 1. More importantly, the non-zero entries of y implies variables and constraints which are important for the infeasibility. Two important features: If y is a certificate of infeasibility, then so is γy, γ > 0. This, however, still identifies the same variables and constraints. An LP can have more than one certificate of infeasibility.. the 25

27 Continued Application of ẋ 1 = 2x x 1x 2 0.5x 1 ẋ 2 = 0.25x 1 x x 1 x x x 2 Using B = [ 3, 3] 2 and d V = 2, the resulting LP is infeasible. Applying once identifies the constraints I 1 = {39, 40, 41, 44, 45}. Setting the corresponding rows of A equal to zero, the reduced problem is also infeasible. Applying on the reduced problem identifies the constraints I 2 = {7, 8, 11, 12, 15}. Setting the corresponding rows of A equal to zero, the second reduced problem is feasible. Thus the union I = I i = {7, 8, 11, 12, 15, 39, 40, 41, 44, 45} i=1,2 contains all the constraints responsible for the infeasibility of the LP. the 26

28 x2 Identified s the x1

29 Can the infeasibility certificate be "generalised" to find all constraints at once? How should the information be utilized? What is the "best" way of sub-dividing? Is there any structure in the problem? Actually yes, the A matrix can be set on Dual Block Angular Structure. Dedicated solver? Benchmark Study comparing complexities with other methods. the

30 Thank you for your attention - Questions?