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Optimization Methods

Optimization Methods. Unconstrained optimization of an objective function F Deterministic, gradient-based methods Running a PDE: will cover later in course Gradient-based (ascent/descent) methods Stochastic methods Simulated annealing Theoretically but not practically interesting

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Optimization Methods

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  1. Optimization Methods • Unconstrained optimization of an objective function F • Deterministic, gradient-based methods • Running a PDE: will cover later in course • Gradient-based (ascent/descent) methods • Stochastic methods • Simulated annealing • Theoretically but not practically interesting • Evolutionary (genetic) algorithms • Multiscale methods • Mean field annealing, graduated nonconvexity, etc. • Constrained optimization • Lagrange multipliers

  2. Our Assumptions for Optimization Methods • With objective function F(p) • Dimension(p) >> 1and frequently quite large • Evaluating F at any p is very expensive • Evaluating D1F at any p is very, very expensive • Evaluating D2F at any p is extremely expensive • True in most image analysis and graphics applications

  3. Order of Convergencefor Iterative Methods • |ei+1| = k| ei|a in limit • a is order of convergence • The major factor in speed of convergence • N steps of method has order of convergence aN • Thus issue is linear convergence (a=1) vs. superlinear convergence(a>1)

  4. Ascent/Descent Methods • At maximum, D1F (i.e., F) =0. • Pick direction of ascent/descent • Find approximate maximum in that direction: two possibilities • Calculate stepsize that will approximately reach maximum • In search direction, find actual max within some range

  5. Gradient Ascent/Descent Methods • Direction of ascent/descent is D1F. • If you move to optimum in that direction, next direction will be orthogonal to this one • Guarantees zigzag • Bad behavior for narrow ridges (valleys) of F • Linear convergence

  6. Newton and Secant Ascent/Descent Methods for F(p) • We are solving D1F=0 • Use Newton or secant equation solution method to solve • Newton to solve f(p)=0 is pi+1 = pi – D1f(pi)-1pi • Newton • Move from p to p-(D2F)-1D1F • Is direction of ascent/descent is gradient direction D1F? • Methods that ascend/descend in D1f (gradient) directionare inferior • Really direction of ascent/descent is direction of (D2F)-1D1F • Also gives you step size in that direction • Secant • Same as Newton except replace D2F and D1F by discrete approximations to them from this and last n iterates

  7. Conjugate gradient method • Preferable to gradient descent/ascent methods • Two major aspects • Successive directions for descent/ascent are conjugate: <hi+1,D2Fhi> = 0 in limit for convex F • If true at all steps (quadratic F), convergence in n-1 steps, with n=dim(p) Improvements available using more previous directions • In search direction, find actual max/min within some range • Quadratic convergence depends on <D1F(xi), hi> =0, i.e., F a local minimum in the hi direction • References • Shewchuk, An Intro. to the CGM w/o the Agonizing Pain (http://www-2.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf) • Numerical Recipes • Polak, Computational Methods in Optimization, Ac. Press

  8. Conjugate gradient method issues • Preferable to gradient descent/ascent methods • Must find a local minimum in the search direction • Will have trouble with • Bumpy objective functions • Extremely elongated minimum/maximum regions

  9. Multiscale Gradient-Based OptimizationTo avoid local optima • Smooth objective function to put initial estimate on hillside of its global optimum • E.g., by using larger scale measurements • Find its optimum • Iterate • Decrease scale of objective function • Use prev. optimum as starting point for new optimization

  10. Multiscale Gradient-Based OptimizationExample Methods • General methods • Graduated non-convexity • [Blake & Zisserman, 1987] • Mean field annealing • [Bilbro, Snyder, et al, 1992] • In image analysis • Vary degree of globality of geometric representation

  11. Optimization under Constraints by Lagrange Multiplier(s) • To optimize F(p) over p subject to gi(p)=0, i=1, 2, …, N, with p having n parameters • Create function F(p)+i li gi(p) • Find critical point for it over p and l • Solve D1p,l[F(p)+i li gi(p)]=0 • n+N equations in n+N unknowns • N of the equations are just gi(p)=0, i=1, 2, …, N • The critical point will need to be an optimum w.r.t. p

  12. Stochastic Methods • Needed when objective function is bumpy or many variables or hard to compute gradient of objective function

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