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MITBoydintro

MITBoydintro - Convex Optimization — Boyd&...

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Unformatted text preview: 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 i ( x ) ≤ b i , i = 1 ,...,m • x = ( x 1 ,...,x n ) : optimization variables • f : R n → R : objective function • f i : R n → 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 bardbl Ax − b bardbl 2 2 solving least-squares problems...
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MITBoydintro - Convex Optimization — Boyd&...

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