4. Convex optimization problems
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1 Convex Optimization Boyd & Vandenberghe 4. Convex optimization problems optimization problem in standard form convex optimization problems quasiconvex optimization linear optimization quadratic optimization geometric programming generalized inequality constraints semidefinite programming vector optimization 4 1
2 Optimization problem in standard form minimize f 0 (x) subject to f i (x) 0, i =1,...,m h i (x) =0, i =1,...,p x R n is the optimization variable f 0 : R n R is the objective or cost function f i : R n R, i =1,...,m, are the inequality constraint functions h i : R n R are the equality constraint functions optimal value: p =inf{f 0 (x) f i (x) 0, i =1,...,m, h i (x) =0, i =1,...,p} p = if problem is infeasible (no x satisfies the constraints) p = if problem is unbounded below Convex optimization problems 4 2
3 Optimal and locally optimal points x is feasible if x dom f 0 and it satisfies the constraints afeasiblex is optimal if f 0 (x) =p ; X opt is the set of optimal points x is locally optimal if there is an R>0 such that x is optimal for minimize (over z) f 0 (z) subject to f i (z) 0, i =1,...,m, h i (z) =0, i =1,...,p z x 2 R examples (with n =1, m = p =0) f 0 (x) =1/x, dom f 0 = R ++ : p =0, no optimal point f 0 (x) = log x, dom f 0 = R ++ : p = f 0 (x) =x log x, dom f 0 = R ++ : p = 1/e, x =1/e is optimal f 0 (x) =x 3 3x, p =, local optimum at x =1 Convex optimization problems 4 3
4 Implicit constraints the standard form optimization problem has an implicit constraint x D = m i=0 dom f i p i=1 dom h i, we call D the domain of the problem the constraints f i (x) 0, h i (x) =0are the explicit constraints a problem is unconstrained if it has no explicit constraints (m = p =0) example: minimize f 0 (x) = k i=1 log(b i a T i x) is an unconstrained problem with implicit constraints a T i x<b i Convex optimization problems 4 4
5 Feasibility problem find x subject to f i (x) 0, i =1,...,m h i (x) =0, i =1,...,p can be considered a special case of the general problem with f 0 (x) =0: minimize 0 subject to f i (x) 0, i =1,...,m h i (x) =0, i =1,...,p p =0if constraints are feasible; any feasible x is optimal p = if constraints are infeasible Convex optimization problems 4 5
6 Convex optimization problem standard form convex optimization problem minimize f 0 (x) subject to f i (x) 0, i =1,...,m a T i x = b i, i =1,...,p f 0, f 1,...,f m are convex; equality constraints are affine problem is quasiconvex if f 0 is quasiconvex (and f 1,...,f m convex) often written as minimize f 0 (x) subject to f i (x) 0, i =1,...,m Ax = b important property: feasible set of a convex optimization problem is convex Convex optimization problems 4 6
7 example minimize f 0 (x) =x x 2 2 subject to f 1 (x) =x 1 /(1 + x 2 2) 0 h 1 (x) =(x 1 + x 2 ) 2 =0 f 0 is convex; feasible set {(x 1,x 2 ) x 1 = x 2 0} is convex not a convex problem (according to our definition): f 1 is not convex, h 1 is not affine equivalent (but not identical) to the convex problem minimize x x 2 2 subject to x 1 0 x 1 + x 2 =0 Convex optimization problems 4 7
8 Local and global optima any locally optimal point of a convex problem is (globally) optimal proof: suppose x is locally optimal, but there exists a feasible y with f 0 (y) <f 0 (x) x locally optimal means there is an R>0 such that z feasible, z x 2 R = f 0 (z) f 0 (x) consider z = θy +(1 θ)x with θ = R/(2 y x 2 ) y x 2 >R,so0 < θ < 1/2 z is a convex combination of two feasible points, hence also feasible z x 2 = R/2 and f 0 (z) θf 0 (y)+(1 θ)f 0 (x) <f 0 (x) which contradicts our assumption that x is locally optimal Convex optimization problems 4 8
9 Optimality criterion for differentiable f 0 x is optimal if and only if it is feasible and f 0 (x) T (y x) 0 for all feasible y X x f 0 (x) if nonzero, f 0 (x) defines a supporting hyperplane to feasible set X at x Convex optimization problems 4 9
10 unconstrained problem: x is optimal if and only if x dom f 0, f 0 (x) =0 equality constrained problem minimize f 0 (x) subject to Ax = b x is optimal if and only if there exists a ν such that x dom f 0, Ax = b, f 0 (x)+a T ν =0 minimization over nonnegative orthant x is optimal if and only if minimize f 0 (x) subject to x 0 x dom f 0, x 0, { f0 (x) i 0 x i =0 f 0 (x) i =0 x i > 0 Convex optimization problems 4 10
11 Equivalent convex problems two problems are (informally) equivalent if the solution of one is readily obtained from the solution of the other, and vice-versa some common transformations that preserve convexity: eliminating equality constraints is equivalent to minimize f 0 (x) subject to f i (x) 0, i =1,...,m Ax = b minimize (over z) f 0 (Fz + x 0 ) subject to f i (Fz + x 0 ) 0, i =1,...,m where F and x 0 are such that Ax = b x = Fz + x 0 for some z Convex optimization problems 4 11
12 introducing equality constraints is equivalent to minimize f 0 (A 0 x + b 0 ) subject to f i (A i x + b i ) 0, i =1,...,m minimize (over x, y i ) f 0 (y 0 ) subject to f i (y i ) 0, i =1,...,m y i = A i x + b i, i =0, 1,...,m introducing slack variables for linear inequalities is equivalent to minimize f 0 (x) subject to a T i x b i, i =1,...,m minimize (over x, s) f 0 (x) subject to a T i x + s i = b i, i =1,...,m s i 0, i =1,...m Convex optimization problems 4 12
13 epigraph form: standard form convex problem is equivalent to minimize (over x, t) t subject to f 0 (x) t 0 f i (x) 0, i =1,...,m Ax = b minimizing over some variables is equivalent to where f 0 (x 1 )=inf x2 f 0 (x 1,x 2 ) minimize f 0 (x 1,x 2 ) subject to f i (x 1 ) 0, i =1,...,m minimize f0 (x 1 ) subject to f i (x 1 ) 0, i =1,...,m Convex optimization problems 4 13
14 Linear program (LP) minimize subject to c T x + d Gx h Ax = b convex problem with affine objective and constraint functions feasible set is a polyhedron c P x Convex optimization problems 4 17
15 Examples diet problem: choose quantities x 1,...,x n of n foods one unit of food j costs c j, contains amount a ij of nutrient i healthy diet requires nutrient i in quantity at least b i to find cheapest healthy diet, minimize c T x subject to Ax b, x 0 piecewise-linear minimization minimize max i=1,...,m (a T i x + b i) equivalent to an LP minimize t subject to a T i x + b i t, i =1,...,m Convex optimization problems 4 18
16 Chebyshev center of a polyhedron Chebyshev center of P = {x a T i x b i, i =1,...,m} is center of largest inscribed ball x cheb B = {x c + u u 2 r} a T i x b i for all x B if and only if sup{a T i (x c + u) u 2 r} = a T i x c + r a i 2 b i hence, x c, r can be determined by solving the LP maximize r subject to a T i x c + r a i 2 b i, i =1,...,m Convex optimization problems 4 19
17 Quadratic program (QP) minimize subject to (1/2)x T Px+ q T x + r Gx h Ax = b P S n +, so objective is convex quadratic minimize a convex quadratic function over a polyhedron f 0 (x ) x P Convex optimization problems 4 22
18 Examples least-squares minimize Ax b 2 2 analytical solution x = A b (A is pseudo-inverse) can add linear constraints, e.g., l x u linear program with random cost minimize c T x + γx T Σx = E c T x + γ var(c T x) subject to Gx h, Ax = b c is random vector with mean c and covariance Σ hence, c T x is random variable with mean c T x and variance x T Σx γ > 0 is risk aversion parameter; controls the trade-off between expected cost and variance (risk) Convex optimization problems 4 23
19 Quasiconvex optimization minimize f 0 (x) subject to f i (x) 0, i =1,...,m Ax = b with f 0 : R n R quasiconvex, f 1,...,f m convex can have locally optimal points that are not (globally) optimal (x, f 0 (x)) Convex optimization problems 4 14
20 convex representation of sublevel sets of f 0 if f 0 is quasiconvex, there exists a family of functions φ t such that: φ t (x) is convex in x for fixed t t-sublevel set of f 0 is 0-sublevel set of φ t, i.e., f 0 (x) t φ t (x) 0 example f 0 (x) = p(x) q(x) with p convex, q concave, and p(x) 0, q(x) > 0 on dom f 0 can take φ t (x) =p(x) tq(x): for t 0, φ t convex in x p(x)/q(x) t if and only if φ t (x) 0 Convex optimization problems 4 15
21 quasiconvex optimization via convex feasibility problems φ t (x) 0, f i (x) 0, i =1,...,m, Ax= b (1) for fixed t, a convex feasibility problem in x if feasible, we can conclude that t p ;ifinfeasible,t p Bisection method for quasiconvex optimization given l p, u p,toleranceϵ > 0. repeat 1. t := (l + u)/2. 2. Solve the convex feasibility problem (1). 3. if (1) is feasible, u := t; else l := t. until u l ϵ. requires exactly log 2 ((u l)/ϵ) iterations (where u, l are initial values) Convex optimization problems 4 16
22 Linear-fractional program minimize subject to f 0 (x) Gx h Ax = b linear-fractional program f 0 (x) = ct x + d e T x + f, dom f 0(x) ={x e T x + f>0} a quasiconvex optimization problem; can be solved by bisection also equivalent to the LP (variables y, z) minimize subject to c T y + dz Gy hz Ay = bz e T y + fz =1 z 0 Convex optimization problems 4 20
23 Second-order cone programming minimize f T x subject to A i x + b i 2 c T i x + d i, i =1,...,m Fx = g (A i R n i n, F R p n ) inequalities are called second-order cone (SOC) constraints: (A i x + b i,c T i x + d i ) second-order cone in R n i+1 for n i =0,reducestoanLP;ifc i =0, reduces to a QCQP more general than QCQP and LP Convex optimization problems 4 25
24 Robust linear programming the parameters in optimization problems are often uncertain, e.g., in an LP there can be uncertainty in c, a i, b i minimize c T x subject to a T i x b i, i =1,...,m, two common approaches to handling uncertainty (in a i,forsimplicity) deterministic model: constraints must hold for all a i E i minimize c T x subject to a T i x b i for all a i E i, i =1,...,m, stochastic model: a i is random variable; constraints must hold with probability η minimize c T x subject to prob(a T i x b i) η, i =1,...,m Convex optimization problems 4 26
25 deterministic approach via SOCP choose an ellipsoid as E i : E i = {ā i + P i u u 2 1} (ā i R n, P i R n n ) center is ā i, semi-axes determined by singular values/vectors of P i robust LP minimize c T x subject to a T i x b i a i E i, i =1,...,m is equivalent to the SOCP minimize c T x subject to ā T i x + P i T x 2 b i, i =1,...,m (follows from sup u 2 1(ā i + P i u) T x =ā T i x + P i T x 2) Convex optimization problems 4 27
26 stochastic approach via SOCP assume a i is Gaussian with mean ā i, covariance Σ i (a i N (ā i, Σ i )) a T i x is Gaussian r.v. with mean āt i x,variancext Σ i x;hence ( ) prob(a T b i ā T i i x b i )=Φ x Σ 1/2 i x 2 where Φ(x) =(1/ 2π) x e t2 /2 dt is CDF of N (0, 1) robust LP minimize c T x subject to prob(a T i x b i) η, i =1,...,m, with η 1/2, is equivalent to the SOCP minimize c T x subject to ā T i x + Φ 1 (η) Σ 1/2 i x 2 b i, i =1,...,m Convex optimization problems 4 28
27 Geometric programming monomial function f(x) =cx a 1 1 xa 2 2 xa n n, dom f = R n ++ with c>0; exponent a i can be any real number posynomial function: sum of monomials f(x) = K k=1 c k x a 1k 1 xa 2k 2 x a nk n, dom f = R n ++ geometric program (GP) with f i posynomial, h i monomial minimize f 0 (x) subject to f i (x) 1, i =1,...,m h i (x) =1, i =1,...,p Convex optimization problems 4 29
28 Geometric program in convex form change variables to y i = log x i, and take logarithm of cost, constraints monomial f(x) =cx a 1 1 xa n n transforms to log f(e y 1,...,e y n )=a T y + b (b = log c) posynomial f(x) = K k=1 c kx a 1k 1 xa 2k 2 x a nk n ( K log f(e y 1,...,e y n ) = log k=1 e at k y+b k transforms to ) (b k = log c k ) geometric program transforms to convex problem ( K ) minimize log k=1 exp(at 0k y + b 0k) ( K ) subject to log k=1 exp(at ik y + b ik) 0, Gy + d =0 i =1,...,m Convex optimization problems 4 30
29 Generalized inequality constraints convex problem with generalized inequality constraints minimize f 0 (x) subject to f i (x) Ki 0, i =1,...,m Ax = b f 0 : R n R convex; f i : R n R k i K i -convex w.r.t. proper cone K i same properties as standard convex problem (convex feasible set,local optimum is global, etc.) conic form problem: special case with affine objective and constraints minimize c T x subject to Fx+ g K 0 Ax = b extends linear programming (K = R m +) to nonpolyhedral cones Convex optimization problems 4 35
30 Semidefinite program (SDP) with F i, G S k minimize c T x subject to x 1 F 1 + x 2 F x n F n + G 0 Ax = b inequality constraint is called linear matrix inequality (LMI) includes problems with multiple LMI constraints: for example, x 1 ˆF1 + + x n ˆFn + Ĝ 0, x 1 F x n Fn + G 0 is equivalent to single LMI x 1 [ ˆF1 0 0 F1 ] +x 2 [ ˆF2 0 0 F2 ] + +x n [ ˆFn 0 0 Fn ] [ Ĝ G ] 0 Convex optimization problems 4 36
31 Eigenvalue minimization minimize λ max (A(x)) where A(x) =A 0 + x 1 A x n A n (with given A i S k ) equivalent SDP minimize subject to t A(x) ti variables x R n, t R follows from λ max (A) t A ti Convex optimization problems 4 38
32 Matrix norm minimization minimize A(x) 2 = ( λ max (A(x) T A(x)) ) 1/2 where A(x) =A 0 + x 1 A x n A n (with given A i R p q ) equivalent SDP minimize subject to t[ ti A(x) T A(x) ti ] 0 variables x R n, t R constraint follows from A 2 t A T A t 2 I, t 0 [ ] ti A A T 0 ti Convex optimization problems 4 39
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