Optimization Problems in Rn Parametric Form Examples The objectives of Optimization Theory

Mathematical Economics - Part I Optimization

Filomena Garcia

Existence of Solutions Unconstrained Optima Equality Constraints Inequality Constraints Convex Structures in Optimization Theory Quasiconvexity

Fall 2009

Filomena Garcia

Optimization

Optimization Filomena Garcia Optimization

Optimization Problems in Rn Parametric Form Examples The objectives of Optimization Theory

1 Optimization in Rn

Optimization Problems in Rn Optimization Problems in Parametric Form Optimization Problems: Some Examples The objectives of Optimization Theory 2 Existence of Solutions 3 Unconstrained Optima 4 Equality Constraints 5 Inequality Constraints 6 Convex Structures in Optimization Theory 7 Quasiconvexity in Optimization 8 Parametric Continuity: The Maximum Theorem 9 Supermodularity and Parametric Monotonicity Filomena Garcia Optimization

Existence of Solutions Unconstrained Optima Equality Constraints Inequality Constraints Convex Structures in Optimization Theory Quasiconvexity

Optimization Problems in Rn

Optimization Filomena Garcia Optimization

Optimization Problems in Rn Parametric Form Examples The objectives of Optimization Theory

An optimization problem in Rn is one where the values of a given function f : Rn → R are to be maximized or minimized over a given set D ⊂ Rn . The function f is called objective function and the set D is called the constraint set. We denote the optimization problem as: max {f (x) |x ∈ D} A solution to the problem is a point x ∈ D such that f (x) ≥ f (y ) for all y ∈ D. We call f (D) the set of attainable values of f in D.

Existence of Solutions Unconstrained Optima Equality Constraints Inequality Constraints Convex Structures in Optimization Theory Quasiconvexity

Filomena Garcia

Optimization

Optimization Filomena Garcia Optimization

Optimization Problems in Rn Parametric Form Examples The objectives of Optimization Theory

It is worth noting the following: 1 A solution to an optimization problem may not exist Example: Let D = R+ and f (x) = x, then f (D) = R+ and supf (D) = +∞, so the problem max {f (x)|x ∈ D} has no solution. 2

Existence of Solutions Unconstrained Optima Equality Constraints Inequality Constraints Convex Structures in Optimization Theory Quasiconvexity

There may be multiple solutions to the optimization problem

Example: Let D = [−1, 1] and f (x) = x 2 , then the maximization problem max {f (x)|x ∈ D} has two solutions: x = 1 and x = −1.

Filomena Garcia

Optimization

Optimization Filomena Garcia Optimization

Optimization Problems in Rn Parametric Form Examples The objectives of Optimization Theory

We will be concerned about the set of solutions to the optimization problem, knowing that this set could be empty. argmax [f (x)|x ∈ D] = {x ∈ D |f (x) ≥ f (y ) , ∀y ∈ D } Two important results to bear in mind: 1

Existence of Solutions Unconstrained Optima Equality Constraints Inequality Constraints Convex Structures in Optimization Theory Quasiconvexity

x is a maximum of f on D if and only if it is a minimum of −f on D. Let ϕ : R → R be a strictly increasing function. Then x is a maximum of f on D if and only if x is also a maximum of the composition ϕ ◦ f .

2

Filomena Garcia

Optimization

Optimization Problems in Parametric Form

Optimization Filomena Garcia Optimization

Optimization Problems in Rn Parametric Form Examples The objectives of Optimization Theory

Optimization problems are often presented in parametric form, i.e. both the objective function and/or the feasible set depend on some parameter θ from a set of feasible parameter values Θ. In this case, we denote the optimization problem as: max {f (x, θ)|x ∈ D(θ)} In general, this problem has a solution which also depends on θ, i.e. x(θ) = argmax {f (x, θ)|x ∈ D(θ)}

Existence of Solutions Unconstrained...