|LEC #||TOPICS||LECTURE NOTES|
Mathematical optimization; least-squares and linear programming; convex optimization; course goals and topics; nonlinear optimization.
Convex sets and cones; some common and important examples; operations that preserve convexity.
Convex functions; common examples; operations that preserve convexity; quasiconvex and log-convex functions.
Convex optimization problems
Convex optimization problems; linear and quadratic programs; second-order cone and semidefinite programs; quasiconvex optimization problems; vector and multicriterion optimization.
Lagrange dual function and problem; examples and applications.
Approximation and fittingNorm approximation; regularization; robust optimization.
Maximum likelihood and MAP estimation; detector design; experiment design.
Projection; extremal volume ellipsoids; centering; classification; placement and location problems.
Filter design and equalization
FIR filters; general and symmetric lowpass filter design; Chebyshev equalization; magnitude design via spectral factorization.
Multi-period processor speed scheduling; minimum time optimal control; grasp force optimization; optimal broadcast transmitter power allocation; phased-array antenna beamforming; optimal receiver location.
l1 methods for convex-cardinality problems
Convex-cardinality problems and examples; l1 heuristic; interpretation as relaxation.
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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Stochastic programming; "certainty equivalent" problem; violation/shortfall constraints and penalties; Monte Carlo sampling methods; validation.
Chance constrained optimization
Chance constraints and percentile optimization; chance constraints for log-concave distributions; convex approximation of chance constraints.
Numerical linear algebra background
Basic linear algebra operations; factor-solve methods; sparse matrix methods.
Gradient and steepest descent methods; Newton method; self-concordance complexity analysis.
Equality constrained minimization
Elimination method; Newton method; infeasible Newton method.
Barrier method; sequential unconstrained minimization; self-concordance complexity analysis.
Disciplined convex programming and CVX
Convex optimization solvers; modeling systems; disciplined convex programming; CVX.