# Computacion Inteligente Derivative-Based Optimization

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• ## 3 basic ingredients…

• an objective function,
• a set of decision variables,
• a set of equality/inequality constraints.

## Design Variables: decision and objective vector

• Design Variables: decision and objective vector
• Constraints: equality and inequality
• Bounds: feasible ranges for variables
• Objective Function: maximization can be converted to minimization due to the duality principle

• ## Objective functions may be unimodal or multimodal.

• Unimodal – only one optimum
• Multimodal – more than one optimum

• ## If f(x) is continuous on the feasible set S which is closed and bounded, then f(x) has a global minimum in S

• A set S is closed if it contains all its boundary pts.
• A set S is bounded if it is contained in the interior of some circle

• ## Derivative-based optimization (gradient based)

• Capable of determining “search directions” according to an objective function’s derivative information
• steepest descent method;
• Newton’s method; Newton-Raphson method;
• ## Derivative-free optimization

• random search method;
• genetic algorithm;
• simulated annealing; etc.

• ## A symmetric matrix M = MT is positive definite if and only if its eigenvalues λi > 0. (semidefinite ↔ λi ≥ 0)

• Proof (): Let vi the eigenvector for the i-th eigenvalue λi
• Then,
• which implies λi > 0,

• ## f: Rn → R is a quadratic function if

• where Q is symmetric.

• ## It is no necessary for Q be symmetric.

• Suposse matrix P non-symmetric

## Suposse matrix P non-symmetric. Example

• Suposse matrix P non-symmetric. Example

• ## Two other shapes can result from the quadratic form.

• If Q is negative definite, then f is a parabolic “bowl” up side down.
• If Q is indefinite then f describes a saddle.

• ## Quadratics are useful in the study of optimization.

• Often, objective functions are “close to” quadratic near the solution.
• It is easier to analyze the behavior of algorithms when applied to quadratics.
• Analysis of algorithms for quadratics gives insight into their behavior in general.

• ## Proof:

• Assign:
• by chain rule:

• ## Proof:

• On the other hand for general v:

• ## If the derivative of ∇f exists, we say that f is twice differentiable.

• Write the second derivative as D2f (or F), and call it the Hessian of f.

• ## Proof of fact:

• Imagine a particle traveling along the level set.
• Let g(t) be the position of the particle at time t, with g(0) = x0.
• Note that f(g(t)) = constant for all t.
• Velocity vector g′(t) is tangent to the level set.
• Consider F(t) = f(g(t)). We have F′(0) = 0. By the chain rule,
• Hence, ∇f(x0) and g′(0) are orthogonal.

• ## Suppose f: Rn → R.

• If f in C1, then
• If f in C2, then

• ## We already know that ∇f(x0) is orthogonal to the level set at x0.

• Suppose ∇f(x0) ≠ 0.

• ## Consider xα = x0 + α∇f(x0), α > 0.

• By Taylor's formula,

• ## The Theorem:

• Suppose f: Rn R C1 smooth, and exist continuous function: k: Rn → [0,1], and,
• And, the search vectors constructed by the model algorithm satisfy:

• And
• ## Then

• if is the sequence constructed by the algorithm model,
• then any accumulation point y of this sequence satisfy:

• ## Theorem:

• If is a sequence constructed by the SD algorithm, then every accumulation point y of the sequence satisfy:
• Proof: from Wolfe theorem

• ## How long a step to take?

• From the chain rule:

## Cada elipsoide tiene f(x) constante

• Cada elipsoide tiene f(x) constante

• ## What is the problem with steepest descent?

• We can repeat the same directions over and over…

• ## What is the problem with steepest descent?

• We can repeat the same directions over and over…

• That is

• ## How do we find ?

• Since search vectors form a basis

• ## We want that after n step the error will be 0:

• Here an idea: if then:

• ## So we look for such that

• Simple calculation shows that if we take

• ## Gordon C. Everstine, Selected Topics in Linear Algebra. The GeorgeWashington University. 8 June 2004

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