Introduction to unconstrained optimization - direct search methods

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1 Introduction to unconstrained optimization - direct search methods Jussi Hakanen Post-doctoral researcher jussi.hakanen@jyu.fi

2 Structure of optimization methods Typically Constraint handling converts the problem to (a series of) unconstrained problems In unconstrained optimization a search direction is determined at each iteration The best solution in the search direction is found with line search Constraint handling method Unconstrained optimization Line search

3 Group discussion 1. What kind of optimality conditions there exist for unconstrained optimization (x R n )? 2. List methods for unconstrained optimization? what are their general ideas? Discuss in small groups (3-4) for minutes Each group has a secretary who writes down the answers of the group At the end, we summarize what each group found

4 Reminder: gradient and hessian Definition: If function f: R n R is differentiable, then the gradient f(x) consists of the partial derivatives f(x) i.e. f x = f(x),, f(x) x 1 x n Definition: If f is twice differentiable, then the matrix 2 f(x) 2 f(x) x 1 x 1 x 1 x n H x = 2 f(x) 2 f(x) x n x 1 x n x n is called the Hessian (matrix) of f at x Result: If f is twice continuously differentiable, then 2 f(x) x i x j = 2 f(x) x j x i T x i

5 Reminder: Definite Matrices Definition: A symmetric n n matrix H is positive semidefinite if x R n x T Hx 0. Definition: A symmetric n n matrix H is positive definite if x T Hx > 0 0 x R n Note: If (> <), then H is negative semidefinite (definite). If H is neither positive nor negative semidefinite, then it is indefinite. Result: Let S R n be open convex set and f: S R twice differentiable in S. Function f is convex if and only if H(x ) is positive semidefinite for all x S.

6 Unconstraint problem min f x, s. t. x R n Necessary conditions: Let f be twice differentiable in x. If x is a local minimizer, then f x = 0 (that is, x is a critical point of f) and H x is positive semidefinite. Sufficient conditions: Let f be twice differentiable in x. If f x = 0 and H(x ) is positive definite, then x is a strict local minimizer. Result: Let f: R n R is twice differentiable in x. If f x = 0 and H(x ) is indefinite, then x is a saddle point.

7 Unconstraint problem Adopted from Prof. L.T. Biegler (Carnegie Mellon University)

8 Descent direction Definition: Let f: R n R. A vector d R n is a descent direction for f in x R n if δ > 0 s.t. f x + λd < f(x ) λ (0, δ]. Result: Let f: R n R be differentiable in x. If d R n s.t. f x T d < 0 then d is a descent direction for f in x.

9 Model algorithm for unconstrained minimization Let x h be the current estimate for x 1) [Test for convergence.] If conditions are satisfied, stop. The solution is x h. 2) [Compute a search direction.] Compute a nonzero vector d h R n which is the search direction. 3) [Compute a step length.] Compute α h > 0, the step length, for which it holds that f x h + α h d h < f(x h ). 4) [Update the estimate for minimum.] Set x h+1 = x h + α h d h, h = h + 1 and go to step 1. From Gill et al., Practical Optimization, 1981, Academic Press

10 On convergence Iterative method: a sequence {x h } s.t. x h x when h Definition: A method converges linearly if α [0,1) and M 0 s.t. h M x h+1 x α x h x, superlinearly if M 0 and for some sequence α h 0 it holds that h M x h+1 x α h x h x, with degree p if α 0, p > 0 and M 0 s.t. h M x h+1 x α x h x p. If p = 2 (p = 3), the convergence is quadratic (cubic).

11 Summary of group discussion for methods 1. Newton s method 1. Utilizes tangent 2. Golden section method 1. For line search 3. Downhill Simplex 4. Cyclic coordinate method 1. One coordinate at a time 5. Polytopy search (Nelder-Mead) 1. Idea based on geometry 6. Gradient descent (steepest descent) 1. Based on gradient information

12 Direct search methods Univariate search, coordinate descent, cyclic coordinate search Hooke and Jeeves Powell s method

13 From Miettinen: Nonlinear optimization, 2007 (in Finnish) Coordinate descent f x = 2x x 1 x 2 + x x 1 x 2

14 From Miettinen: Nonlinear optimization, 2007 (in Finnish) Idea of pattern search

15 From Miettinen: Nonlinear optimization, 2007 (in Finnish) Hooke and Jeeves f x = x x 1 2x 2 2

16 From Miettinen: Nonlinear optimization, 2007 (in Finnish) Hooke and Jeeves with fixed step length f x = x x 1 2x 2 2

17 Powell s method Most efficient pattern search method Differs from Hooke and Jeeves so that for each pattern search step one of the coordinate directions is replaced with previous pattern search direction.

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