07-01-Convex-Optimization.pdf

# 07-01-Convex-Optimization.pdf - Convex Optimization CSE 440...

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Convex Optimization CSE 440

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Today I Optimization Setup I Convex Optimization
Optimization Setup

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Optimization Problem: Definition I Optimization Problem: Determine value of optimization variable within feasible region/set to optimize optimization objective min x f ( x ) s.t. x ∈ F I Optimization variable x R n I Feasible region/set F ⊂ R n I Optimization objective: f : F → R
Optimization Problem: Definition I Optimization Problem: Determine value of optimization variable within feasible region/set to optimize optimization objective min x f ( x ) s.t. x ∈ F I Optimization variable x R n I Feasible region/set F ⊂ R n I Optimization objective: f : F → R I Optimal solution: x * = arg min x ∈F f ( x )

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Optimization Problem: Definition I Optimization Problem: Determine value of optimization variable within feasible region/set to optimize optimization objective min x f ( x ) s.t. x ∈ F I Optimization variable x R n I Feasible region/set F ⊂ R n I Optimization objective: f : F → R I Optimal solution: x * = arg min x ∈F f ( x ) I Optimal objective value f * = min x ∈F f ( x ) = f ( x * )
Optimization Problem: Definition min x f ( x ) s.t. x ∈ F I Optimization variable x R n

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Optimization Problem: Definition min x f ( x ) s.t. x ∈ F I Optimization variable x R n I Discrete variables: Combinatorial optimization I Continuous variables: Continuous optimization I Mixed: Some variables are discrete, and some are continuous
Optimization Problem: Definition min x f ( x ) s.t. x F I Feasible region/set F ⊂ R n I Unconstrained optimization: I Constrained optimization:

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Optimization Problem: Definition min x f ( x ) s.t. x F I Feasible region/set F ⊂ R n I Unconstrained optimization: F = R n I Constrained optimization:
Optimization Problem: Definition min x f ( x ) s.t. x F I Feasible region/set F ⊂ R n I Unconstrained optimization: F = R n I Constrained optimization: F ⊆ R n

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Optimization Problem: Definition min x f ( x ) s.t. x F I Feasible region/set F ⊂ R n I Unconstrained optimization: F = R n I Constrained optimization: F ⊆ R n I Finding a feasible point x ∈ F can already be difficult
Optimization Problem: Definition min x f ( x ) s.t. x ∈ F I Optimization objective f : F → R I f ( x ) = 1 : Feasibility problem

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Optimization Problem: Definition min x f ( x ) s.t. x ∈ F I Optimization objective f : F → R I f ( x ) = 1 : Feasibility problem I Simple functions I Linear function f ( x ) = a T x I Convex function (this lecture)
Optimization Problem: Definition min x f ( x ) s.t. x ∈ F I Optimization objective f : F → R I f ( x ) = 1 : Feasibility problem I Simple functions I Linear function f ( x ) = a T x I Convex function (this lecture) I Complicated functions I Can be implicitly represented through an algorithm which takes x ∈ F as input, and outputs a value

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Optimization Problem: Definition min x f ( x ) s.t. x ∈ F I Minimization can be converted to maximization (and vice versa) max x g ( x ) = - f ( x ) s.t. x ∈ F
• Winter '15
• Optimization, optimization problem, Convex Optimization, optimization setup

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