Gradient Methods April 2004

Preview Background Steepest Descent Conjugate Gradient

Preview Background Steepest Descent Conjugate Gradient

Background Motivation The gradient notion The Wolfe Theorems

Motivation The min(max) problem: But we learned in calculus how to solve that kind of question!

Motivation Not exactly, Functions: High order polynomials: What about function that don ’ t have an analytic presentation: “ Black Box ”

Motivation- “ real world ” problem Connectivity shapes (isenburg,gumhold,gotsman) What do we get only from C without geometry?

Motivation- “ real world ” problem First we introduce error functionals and then try to minimize them:

Motivation- “ real world ” problem Then we minimize: High dimension non-linear problem. The authors use conjugate gradient method which is maybe the most popular optimization technique based on what we ’ ll see here.

Motivation- “ real world ” problem Changing the parameter:

Motivation General problem: find global min(max) This lecture will concentrate on finding local minimum.

Background Motivation The gradient notion The Wolfe Theorems

Directional Derivatives: first, the one dimension derivative:

Directional Derivatives : Along the Axes …

Directional Derivatives : In general direction …

Directional Derivatives

In the plane The Gradient: Definition in

The Gradient: Definition

The Gradient Properties The gradient defines (hyper) plane approximating the function infinitesimally

The Gradient properties By the chain rule: (important for later use)

The Gradient properties Proposition 1: is maximal choosing is minimal choosing (intuitive: the gradient point the greatest change direction)

The Gradient properties Proof: (only for minimum case) Assign: by chain rule:

The Gradient properties On the other hand for general v:

The Gradient Properties Proposition 2: let be a smooth function around P, if f has local minimum (maximum) at p then, (Intuitive: necessary for local min(max))

The Gradient Properties Proof: Intuitive:

The Gradient Properties Formally: for any We get:

The Gradient Properties We found the best INFINITESIMAL DIRECTION at each point, Looking for minimum: “ blind man ” procedure How can we derive the way to the minimum using this knowledge?

Background Motivation The gradient notion The Wolfe Theorems

The Wolfe Theorem This is the link from the previous gradient properties to the constructive algorithm. The problem:

The Wolfe Theorem We introduce a model for algorithm: Data: Step 0:set i=0 Step 1:ifstop, else, compute search direction Step 2: compute the step-size Step 3:setgo to step 1

The Wolfe Theorem The Theorem: suppose C1 smooth, and exist continuous function: And, And, the search vectors constructed by the model algorithm satisfy:

The Wolfe Theorem And Then if is the sequence constructed by the algorithm model, then any accumulation point y of this sequence satisfy:

The Wolfe Theorem The theorem has very intuitive interpretation : Always go in decent direction.

Preview Background Steepest Descent Conjugate Gradient

Steepest Descent What it mean? We now use what we have learned to implement the most basic minimization technique. First we introduce the algorithm, which is a version of the model algorithm. The problem:

Steepest Descent Steepest descent algorithm: Data: Step 0:set i=0 Step 1:ifstop, else, compute search direction Step 2: compute the step-size Step 3:setgo to step 1

Steepest Descent Theorem: if is a sequence constructed by the SD algorithm, then every accumulation point y of the sequence satisfy: Proof: from Wolfe theorem Remark: wolfe theorem gives us numerical stability is the derivatives aren ’ t given (are calculated numerically).

Steepest Descent From the chain rule: Therefore the method of steepest descent looks like this:

Steepest Descent

The steepest descent find critical point and local minimum. Implicit step-size rule Actually we reduced the problem to finding minimum: There are extensions that gives the step size rule in discrete sense. (Armijo)

Steepest Descent Back with our connectivity shapes: the authors solve the 1-dimension problem analytically. They change the spring energy and get a quartic polynomial in x

Preview Background Steepest Descent Conjugate Gradient