Convex Optimization - Boyd & Vandenberghe
2. Convex sets
affine and convex sets some important examples operations that preserve convexity generalized inequalities separating and supporting hyperplanes dual cones and generalized inequalities
21
Affine se
Convex Optimization - Boyd & Vandenberghe
3. Convex functions
basic properties and examples operations that preserve convexity the conjugate function quasiconvex functions log-concave and log-convex functions convexity with respect to generalized inequal
Convex Optimization - Boyd & Vandenberghe
7. Statistical estimation
maximum likelihood estimation optimal detector design experiment design
71
Parametric distribution estimation
distribution estimation problem: estimate probability density p(y) of a ran
Filter design
FIR filters Chebychev design linear phase filter design equalizer design filter magnitude specifications
1
FIR filters
finite impulse response (FIR) filter:
n-1 =0
y(t) =
h u(t - ),
tZ
(sequence) u : Z R is input signal (sequence) y : Z R
1-norm Methods for
Convex-Cardinality Problems
problems involving cardinality the 1-norm heuristic convex relaxation and convex envelope interpretations examples recent results
Prof. S. Boyd, EE364b, Stanford University
1-norm heuristics for cardinality
Convex Optimization - Boyd & Vandenberghe
8. Geometric problems
extremal volume ellipsoids centering classification placement and facility location
81
Minimum volume ellipsoid around a set
Lwner-John ellipsoid of a set C: minimum volume ellipsoid E s.t.
Convex Optimization - Boyd & Vandenberghe
9. Numerical linear algebra background
matrix structure and algorithm complexity solving linear equations with factored matrices LU, Cholesky, LDLT factorization block elimination and the matrix inversion lemma s
6.079/6.975 December 1011, 2009.
S. Boyd & P. Parrilo
Final exam
This is a 24 hour take-home final exam. Please turn it in to Professor Stephen Boyd, (Stata Center), on Friday December 11, at 5PM (or before). You may use any books, notes, or computer prog
6.079/6.975 October 2930, 2009.
S. Boyd & P. Parrilo
Midterm exam
This is a 24 hour take-home midterm exam. Please turn it in to Professor Pablo Parrilo, (Stata Center), on Friday October 30, at 5PM (or before). You may use any books, notes, or computer p
6.079/6.975, Fall 200910
S. Boyd & P. Parrilo
Homework 3 additional problems
1. Reverse Jensen inequality. Suppose f is convex, 1 > 0, i 0, i = 2, . . . , k, and 1 + + n = 1, and let x1 , . . . , xn dom f . Show that the inequality f (1 x1 + + n xn ) 1 f
6.079/6.975, Fall 200910
S. Boyd & P. Parrilo
Homework 4 additional problems
1. Simple portfolio optimization. We consider a portfolio optimization problem as de scribed on pages 155 and 185186 of Convex Optimization, with data that can be found in the fi
6.079/6.975, Fall 200910
S. Boyd & P. Parrilo
Homework 5 additional problems
1. Heuristic suboptimal solution for Boolean LP. This exercise builds on exercises 4.15 and 5.13 in Convex Optimization, which involve the Boolean LP minimize cT x subject to Ax
6.079/6.975, Fall 2009-10
S. Boyd & P. Parrilo
Homework 6 additional problems
1. Maximizing house profit in a gamble and imputed probabilities. A set of n participants bet on which one of m outcomes, labeled 1, . . . , m, will occur. Participant i offers
6.079/6.975, Fall 2009-10
S. Boyd & P. Parrilo
Homework 7 additional problems
1. Identifying a sparse linear dynamical system. A linear dynamical system has the form x(t + 1) = Ax(t) + Bu(t) + w(t), t = 1, . . . , T - 1,
where x(t) Rn is the state, u(t) R
6.079/6.975, Fall 2009-10
S. Boyd & P. Parrilo
Homework 8
1. Conformal mapping via convex optimization. Suppose that is a closed bounded region in C with no holes (i.e., it is simply connected). The Riemann mapping theorem states that there exists a confo
6.079/6.975, Fall 2009-10
S. Boyd & P. Parrilo
Homework 9
1. Efficient numerical method for a regularized least-squares problem. We consider a reg ularized least squares problem with smoothing, minimize
k i=1
(aT x i
- bi ) +
2
n-1 i=1
(xi - xi+1 ) +
2
6.079/6.975, Fall 2009-10
S. Boyd & P. Parrilo
(Part of) Homework 10: Standard form LP barrier method
In the following three exercises, you will implement a barrier method for solving the standard form LP minimize cT x
subject to Ax = b, x 0,
with varia