6.079 | Fall 2009 | Undergraduate

Introduction to Convex Optimization

Lecture Notes

Notes for Lecture 20 are not available on MIT OpenCourseWare.

LEC # TOPICS LECTURE NOTES
1

Introduction

Mathematical optimization; least-squares and linear programming; convex optimization; course goals and topics; nonlinear optimization.

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2

Convex sets

Convex sets and cones; some common and important examples; operations that preserve convexity.

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3

Convex functions

Convex functions; common examples; operations that preserve convexity; quasiconvex and log-convex functions.

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4

Convex optimization problems

Convex optimization problems; linear and quadratic programs; second-order cone and semidefinite programs; quasiconvex optimization problems; vector and multicriterion optimization.

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5

Duality

Lagrange dual function and problem; examples and applications.

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6

Approximation and fitting

Norm approximation; regularization; robust optimization.

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7

Statistical estimation

Maximum likelihood and MAP estimation; detector design; experiment design.

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8

Geometric problems

Projection; extremal volume ellipsoids; centering; classification; placement and location problems.

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9

Filter design and equalization

FIR filters; general and symmetric lowpass filter design; Chebyshev equalization; magnitude design via spectral factorization.

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10

Miscellaneous applications

Multi-period processor speed scheduling; minimum time optimal control; grasp force optimization; optimal broadcast transmitter power allocation; phased-array antenna beamforming; optimal receiver location.

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11

l1 methods for convex-cardinality problems

Convex-cardinality problems and examples; l1 heuristic; interpretation as relaxation.

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12

l1 methods for convex-cardinality problems (cont.)

Total variation reconstruction; iterated re-weighted l1; rank minimization and dual spectral norm heuristic.

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13

Stochastic programming

Stochastic programming; “certainty equivalent” problem; violation/shortfall constraints and penalties; Monte Carlo sampling methods; validation.

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14

Chance constrained optimization

Chance constraints and percentile optimization; chance constraints for log-concave distributions; convex approximation of chance constraints.

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15

Numerical linear algebra background

Basic linear algebra operations; factor-solve methods; sparse matrix methods.

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16

Unconstrained minimization

Gradient and steepest descent methods; Newton method; self-concordance complexity analysis.

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17

Equality constrained minimization

Elimination method; Newton method; infeasible Newton method.

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18

Interior-point methods

Barrier method; sequential unconstrained minimization; self-concordance complexity analysis.

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19

Disciplined convex programming and CVX

Convex optimization solvers; modeling systems; disciplined convex programming; CVX.

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20 Conclusions  

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