Introduction to Optimization Problems and Methods

Transcription

1 Introduction to Optimization Problems and Methods December 10, 2009

2 Outline 1 Linear Optimization Problem Simplex Method 2 3 Cutting Plane Method 4 Discrete Dynamic Programming

3 Problem Simplex Method General and Standard form of Linear Programming problem General Form Standard Form min c T x s.t. ai T x b i i M 1 ai T x b i i M 2 ai T x = b i i M 3 x j 0 j N 1 x j 0 j N 2 min s.t. c T x Ax = b x 0

4 Simplex Method Linear Optimization Problem Simplex Method Polyhedron Geometric Interpretation of Simplex Method If optimal solution exists, it can be achieved on one of the vertices. The algorithm travels along edges of the polyhedron to vertices with higher function value until it achieves optimal value.

5 Problem Simplex Method Pros and Cons of Simplex Method Pros Remarkably efficient in practice, especially when the scale of the problem is small. Cons Worst-case complexity is exponential time. For almost every pivoting rule, there is an exponential worst-case complexity example.

6 One-Dimensional Nonlinear Problem Problem min x [a,b] f (x) Assumption There is a unique local minimum in the interval considered. f (x) may not be differentiable.

7 Bisection Method Linear Optimization 5a 5b X 5

8 Bisection Method X 5 5a 5b In each iteration, the length of the interval becomes half of the previous one. Calculate the value at two more points in each iteration.

9 Golden Section Method

10 Golden Section Method b a = a c = b c = c In each iteration, the length of the interval becomes of the previous one. Only need to calculate one more point in each iteration.

11 Multi-Dimensional Unconstrained Problem Problem min f (x), x R n Assumption f (x) is 2 times differentiable. Questions: How to choose direction? How to choose step length?

12 Steepest Descent Method Choose d i at x i as d i = f (x i ).

13 Steepest Descent Method Choose d i at x i as d i = f (x i ). Since steepest descent direction is a local property, the method is not effective.

14 Conjugate Gradient Method (1) Definition Nonzero vectors u and v are conjugate with respect to positive definite matrix A if u T Av = 0

15 Conjugate Gradient Method (1) Definition Nonzero vectors u and v are conjugate with respect to positive definite matrix A if u T Av = 0 Property If any two of nonzero vectors v 1, v 2,...v n are conjugate in R n, then v 1, v 2,..., v n form a base in R n.

16 Conjugate Gradient Method (2) Consider f (x) = 1 2 x T Ax b T x, where A is positive definite.

17 Conjugate Gradient Method (2) Consider f (x) = 1 2 x T Ax b T x, where A is positive definite. r k = b Ax k is called residual, which is gradient descent.

18 Conjugate Gradient Method (2) Consider f (x) = 1 2 x T Ax b T x, where A is positive definite. r k = b Ax k is called residual, which is gradient descent. Let p k+1 = r k i k p T i Ar k p T i Ap i p i.

19 Conjugate Gradient Method (2) Consider f (x) = 1 2 x T Ax b T x, where A is positive definite. r k = b Ax k is called residual, which is gradient descent. Let p k+1 = r k i k p T i Ar k p T i Ap i p i. x k+1 = x k + α k+1 p k+1, where α k+1 = pt k+1 r k pk+1 T Ap. k+1

20 Conjugate Gradient Method (3) Avoid repeating directions used before. Convergence in finite steps for quadratic objective functions.

21 Newton Method (1) Consider f (x) = 1 2 (x x k) T A(x x k ) b T (x x k ), where A is positive definite. The minimizer is x k+1 = x k + A 1 b.

22 Newton Method (1) Consider f (x) = 1 2 (x x k) T A(x x k ) b T (x x k ), where A is positive definite. The minimizer is x k+1 = x k + A 1 b. The idea of Newton Method is approximating a function locally at x k by its first three terms of Taylor expansion, and set the next iterate be the minimizer of the approximation.

23 Newton Method (2) Pros and Cons Pros Newton s method can often converge remarkably quickly, especially if the iteration begins sufficiently near the optimal solution. A modification Quasi-Newton methods saves computation. Cons If f (x) is not convex function, the Newton s method may sometimes diverge or converge to saddle point and local minimum.

24 Problem Problem min f (x) s.t. g i (x) 0, j = 1, 2,..., m Assumption g i (x), j = 1, 2,..., m are convex function. So feasible region is convex.

25 Interior Penalty Function Method

26 Interior Penalty Function Method Choose penalty function as 1 g j (x), and let φ(x, r k) = f (x) r k m j=1 1 g j (x). Suppose we have an initial feasible solution. Do unconstrained optimization for φ(x, r k ) in each step. let r k 0 as k. When r k is very small, φ approximately equal to the original constrained problem. More often, penalty function log(g j (x)) is used.

27 Cutting Plane Method Integer Programming-Cutting Plane Method Blue lines are the boundary; Red dots are feasible solutions; Assume cost vector is vertically upward; Green line is the cutting plane.

28 Discrete Dynamic Programming A Discrete Dynamic Programming Example There are T + 1 periods, t = 0, 1,..., T. k t : capital in period t; c t : consumption in period t; u(c t ) utility of consuming c t. k t+1 = Ak a t c t, A > 0, 0 < a < 1 Problem T max λ t u(c t ), subject to k t+1 = Akt a c t 0 t=0

29 Discrete Bellman Equation Discrete Dynamic Programming Define value function V T +1 (k) = 0; Bellman Equation V t (k t ) = max(u(c t ) + λv t+1 (k t+1 )) subject to k t+1 = Ak a t c t 0 for t = 0, 1,..., T Bellman Equation can be solved by backward induction.

30 Discrete Dynamic Programming A Continuous Dynamic Programming Example Problem { T min 0 } C[x(t), a(t)] + D[x(T )] where C( ) is the cost rate function and D( ) is the utility at the final state, x(t) is the system state vector, x(0) is given, and a(t) is control vector. The system is also subject to ẋ(t) = F [x(t), u(t)].

31 Discrete Dynamic Programming HJB Equation Define value function V (x, T ) = D(x); V (x(t), t) = min{c(x(t), a)dt + V (x(t + dt), t + dt)} a = min a {C(x(t), a)dt + V (x, t)dt + V (x, t)ẋdt + o(dt 2 )} HJB Equation V (x, t) + min a { V (x, t)f (x, a) + C(x, a)} = 0 V (x, T ) = D(x)

32 Discrete Dynamic Programming Thank you very much for your attention!