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文库首页人工智能机器学习Lecture_slides for Convex Optimization_Stephen Boyd

Lecture_slides for Convex Optimization_Stephen Boyd

slides

Convex

Optimization

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Lecture_slides for Convex Optimization_Stephen Boyd

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Convex Optimization — Boyd & Vandenberghe

1. Introduction

• mathematical optimization

• least-squares and linear programming

• convex optimization

• example

• course goals and topics

• nonlinear optimization

• brief history of convex optimization

1–1

Mathematical optimization

(mathematical) optimization problem

minimize f

(x)

subject to f

(x) ≤ b

, i = 1, . . . , m

• x = (x

, . . . , x

): optimization variables

• f

: R

→ R: objective function

• f

: R

→ R, i = 1, . . . , m: constraint functions

optimal solution x

has smallest value of f

among all vectors that

satisfy the constraints

Introduction 1–2

Examples

portfolio optimization

• variables: amounts invested in diﬀerent assets

• constraints: budget, max./min. investment per asset, minimum return

• objective: overall risk or return variance

device sizing in electronic circuits

• variables: device widths and lengths

• constraints: manufacturing limits, timing requirements, maximum area

• objective: power consumption

data ﬁtting

• variables: model parameters

• constraints: prior information, parameter limits

• objective: measure of misﬁt or prediction error

Introduction 1–3

Solving optimization problems

general optimization problem

• very diﬃcult to solve

• methods involve some compromise, e.g., very long computation time, or

not always ﬁnding the solution

exceptions: certain problem classes can be solved eﬃciently and reliably

• least-squares problems

• linear programming problems

• convex optimization problems

Introduction 1–4

Least-squares

minimize kAx − bk

solving least-squares problems

• analytical solution: x

= (A

−1

• reliable and eﬃcient algorithms and software

• computation time proportional to n

k (A ∈ R

k×n

); less if structured

• a mature technology

using least-squares

• least-squares problems are easy to recognize

• a few standard techniques increase ﬂexibility (e.g., including weights,

adding regularization terms)

Introduction 1–5

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