Description

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
Optimization problem in standard form
minimize f0(x)
subject to fi(x) ≤ 0, hi(x) = 0, i = 1,…,m
i = 1,…,p
• x ∈ Rn is the optimization variable
• f0 : Rn → R is the objective or cost function
• fi : Rn → R, i = 1,…,m, are the inequality constraint functions
• hi : Rn → R are the equality constraint functions optimal value: p⋆ = inf{f0(x) | fi(x) ≤ 0, i = 1,…,m, hi(x) = 0, i = 1,…,p}
• p⋆ = ∞ if problem is infeasible (no x satisfies the constraints)
• p⋆ = −∞ if problem is unbounded below
Optimal and locally optimal points
x is feasible if x ∈ domf0 and it satisfies the constraints a feasible x is optimal if f0(x) = p⋆; Xopt 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) f0(z)
subject to fi(z) ≤ 0, i = 1,…,m, hi(z) = 0, i = 1,…,p kz − xk2 ≤ R
examples (with n = 1, m = p = 0)
• f0(x) = 1/x, domf0 = R++: p⋆ = 0, no optimal point
• f0(x) = −logx, domf0 = R++: p⋆ = −∞
• f0(x) = xlogx, domf0 = R++: p⋆ = −1/e, x = 1/e is optimal
• f0(x) = x3 − 3x, p⋆ = −∞, local optimum at x = 1
Implicit constraints
the standard form optimization problem has an implicit constraint
dom domhi,
• we call D the domain of the problem
• the constraints fi(x) ≤ 0, hi(x) = 0 are the explicit constraints
• a problem is unconstrained if it has no explicit constraints (m = p = 0)
example: minimize
is an unconstrained problem with implicit constraints
Feasibility problem
find subject to
can be considered a special case of the general problem with f0(x) = 0:
minimize 0
subject to fi(x) ≤ 0, i = 1,…,m
hi(x) = 0, i = 1,…,p
• p⋆ = 0 if constraints are feasible; any feasible x is optimal
• p⋆ = ∞ if constraints are infeasible
Convex optimization problem
standard form convex optimization problem
minimize f0(x)
subject to fi(x) ≤ 0, i = 1,…,m
aTi x = bi, i = 1,…,p
• f0, f1, . . . , fm are convex; equality constraints are affine
• problem is quasiconvex if f0 is quasiconvex (and f1, . . . , fm convex) often written as
minimize f0(x)
subject to fi(x) ≤ 0, i = 1,…,m Ax = b
important property: feasible set of a convex optimization problem is convex example
minimize f0(x) = x21 + x22 subject to f1(x) = x1/(1 + x22) ≤ 0
h1(x) = (x1 + x2)2 = 0
• f0 is convex; feasible set {(x1,x2) | x1 = −x2 ≤ 0} is convex
• not a convex problem (according to our definition): f1 is not convex, h1 is not affine
• equivalent (but not identical) to the convex problem
minimize x21 + x22
subject to x1 ≤ 0 x1 + x2 = 0
Local and global optima
any locally optimal point of a convex problem is (globally) optimal proof: suppose x is locally optimal and y is optimal with f0(y) < f0(x) x locally optimal means there is an R > 0 such that z feasible, kz − xk2 ≤ R =⇒ f0(z) ≥ f0(x)
consider z = θy + (1 − θ)x with θ = R/(2ky − xk2)
• ky − xk2 > R, so 0 < θ < 1/2
• z is a convex combination of two feasible points, hence also feasible
• kz − xk2 = R/2 and
f0(z) ≤ θf0(x) + (1 − θ)f0(y) < f0(x)
which contradicts our assumption that x is locally optimal
Optimality criterion for differentiable f0
x is optimal if and only if it is feasible and
∇f0(x)T(y − x) ≥ 0 for all feasible y

if nonzero, ∇f0(x) defines a supporting hyperplane to feasible set X at x
• unconstrained problem: x is optimal if and only if
x ∈ domf0, ∇f0(x) = 0
• equality constrained problem
minimize f0(x) subject to Ax = b
x is optimal if and only if there exists a ν such that
x ∈ domf0, Ax = b, ∇f0(x) + ATν = 0
• minimization over nonnegative orthant
minimize f0(x) subject to
x is optimal if and only if
x ∈ dom ,
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
minimize f0(x)
subject to
is equivalent to fi(x) ≤ 0, i = 1,…,m Ax = b
minimize (over z) f0(Fz + x0)
subject to fi(Fz + x0) ≤ 0, i = 1,…,m
where F and x0 are such that
Ax = b ⇐⇒ x = Fz + x0 for some z
• introducing equality constraints
minimize f0(A0x + b0) subject to fi(Aix + bi) ≤ 0, i = 1,…,m
is equivalent to
minimize (over x, yi) f0(y0)
subject to fi(yi) ≤ 0, i = 1,…,m yi = Aix + bi, i = 0,1,…,m
• introducing slack variables for linear inequalities
minimize f0(x) subject to
is equivalent to
minimize (over x, s) f0(x)
subject to aTi x + si = bi, i = 1,…,m
si ≥ 0, i = 1,…m
• epigraph form: standard form convex problem is equivalent to
minimize (over x, t) t
subject to f0(x) − t ≤ 0
fi(x) ≤ 0, i = 1,…,m Ax = b
• minimizing over some variables
minimize f0(x1,x2)
subject to
is equivalent to fi(x1) ≤ 0, i = 1,…,m
minimize f˜0(x1)
subject to fi(x1) ≤ 0, i = 1,…,m
where f˜0(x1) = infx2 f0(x1,x2)
Quasiconvex optimization
minimize f0(x) subject to fi(x) ≤ 0, i = 1,…,m
Ax = b
with f0 : Rn → R quasiconvex, f1, . . . , fm convex
can have locally optimal points that are not (globally) optimal

convex representation of sublevel sets of f0 if f0 is quasiconvex, there exists a family of functions φt such that:
• φt(x) is convex in x for fixed t
• t-sublevel set of f0 is 0-sublevel set of φt, i.e.,
f0(x) ≤ t ⇐⇒ φt(x) ≤ 0
example

with p convex, q concave, and p(x) ≥ 0, q(x) > 0 on domf0 can take φt(x) = p(x) − tq(x):
• for t ≥ 0, φt convex in x
• p(x)/q(x) ≤ t if and only if φt(x) ≤ 0
quasiconvex optimization via convex feasibility problems
φt(x) ≤ 0, fi(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⋆; if infeasible, t ≤ p⋆

Bisection method for quasiconvex optimization
given l ≤ p⋆, u ≥ p⋆, tolerance ǫ > 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 ≤ ǫ.

requires exactly ⌈log2((u − l)/ǫ)⌉ iterations (where u, l are initial values)
Linear program (LP)
minimize cTx + d subject to
• convex problem with affine objective and constraint functions
• feasible set is a polyhedron

Examples
diet problem: choose quantities x1, . . . , xn of n foods
• one unit of food j costs cj, contains amount aij of nutrient i
• healthy diet requires nutrient i in quantity at least bi to find cheapest healthy diet,
minimize cTx
subject to
piecewise-linear minimization
minimize
equivalent to an LP
minimize t
subject to
Chebyshev center of a polyhedron Chebyshev center of
is center of largest inscribed ball
B = {xc + u | kuk2 ≤ r}
for all x ∈ B if and only if

• hence, xc, r can be determined by solving the LP
maximize r
subject to
(Generalized) linear-fractional program
minimize f0(x) subject to
linear-fractional program
, domf0(x) = {x | eTx + f > 0}
• a quasiconvex optimization problem; can be solved by bisection
• also equivalent to the LP (variables y, z)
minimize cTy + dz
subject to
generalized linear-fractional program
, dom
a quasiconvex optimization problem; can be solved by bisection
example: Von Neumann model of a growing economy
maximize (over x, x+) mini=1,…,n x+i /xi subject to
• x,x+ ∈ Rn: activity levels of n sectors, in current and next period
• (Ax)i, (Bx+)i: produced, resp. consumed, amounts of good i
• x+i /xi: growth rate of sector i allocate activity to maximize growth rate of slowest growing sector
Quadratic program (QP)
minimize (1/2)xTPx + qTx + r
subject to
, so objective is convex quadratic
• minimize a convex quadratic function over a polyhedron

Examples
least-squares
minimize kAx − bk22
• analytical solution x⋆ = A†b (A† is pseudo-inverse)
• can add linear constraints, e.g., linear program with random cost
minimize c¯Tx + γxTΣx = EcTx + γ var(cTx)
subject to
• c is random vector with mean c¯ and covariance Σ
• hence, cTx is random variable with mean c¯Tx and variance xTΣx
• γ > 0 is risk aversion parameter; controls the trade-off between expected cost and variance (risk)
Quadratically constrained quadratic program (QCQP)
minimize
subject to (1/2)xTPix + qiTx + ri ≤ 0, i = 1,…,m Ax = b
; objective and constraints are convex quadratic
, feasible region is intersection of m ellipsoids and
an affine set
Second-order cone programming
minimize fTx
subject to kAix + bik2 ≤ cTi x + di, i = 1,…,m Fx = g
(Ai ∈ Rni×n, F ∈ Rp×n)
• inequalities are called second-order cone (SOC) constraints:
second-order cone in Rni+1
• for ni = 0, reduces to an LP; if ci = 0, reduces to a QCQP
• more general than QCQP and LP
Robust linear programming
the parameters in optimization problems are often uncertain, e.g., in an LP
minimize cTx subject to aTi x ≤ bi, i = 1,…,m,
there can be uncertainty in c, ai, bi
two common approaches to handling uncertainty (in ai, for simplicity)
• deterministic model: constraints must hold for all ai ∈ Ei
minimize cTx
subject to for all ai ∈ Ei, i = 1,…,m,
• stochastic model: ai is random variable; constraints must hold with probability η
minimize cTx subject to prob deterministic approach via SOCP
• choose an ellipsoid as Ei:
Ei = {a¯i + Piu | kuk2 ≤ 1} (¯ai ∈ Rn, Pi ∈ Rn×n)
center is a¯i, semi-axes determined by singular values/vectors of Pi
• robust LP
minimize cTx
subject to
is equivalent to the SOCP
minimize cTx subject to
(follows from
stochastic approach via SOCP
• assume ai is Gaussian with mean a¯i, covariance Σi (ai ∼ N(¯ai,Σi)) is Gaussian r.v. with mean , variance xTΣix; hence prob !
where is CDF of N(0,1)
• robust LP
minimize cTx subject to prob
with η ≥ 1/2, is equivalent to the SOCP
minimize cTx subject to
Geometric programming
monomial function
, dom
with c > 0; exponent αi can be any real number posynomial function: sum of monomials
, dom
geometric program (GP)
minimize f0(x)
subject to fi(x) ≤ 1, hi(x) = 1, i = 1,…,m
i = 1,…,p
with fi posynomial, hi monomial
Geometric program in convex form
change variables to yi = logxi, and take logarithm of cost, constraints
• monomial transforms to
logf(ey1,…,eyn) = aTy + b (b = logc)
• posynomial transforms to
! (bk = logck)
• geometric program transforms to convex problem minimize subject to
Design of cantilever beam
segment 4 segment 3 segment 2 segment 1

F
• N segments with unit lengths, rectangular cross-sections of size wi × hi
• given vertical force F applied at the right end design problem
minimize total weight
subject to upper & lower bounds on wi, hi upper bound & lower bounds on aspect ratios hi/wi upper bound on stress in each segment
upper bound on vertical deflection at the end of the beam
variables: wi, hi for i = 1,…,N
objective and constraint functions
• total weight w1h1 + ··· + wNhN is posynomial
• aspect ratio hi/wi and inverse aspect ratio wi/hi are monomials
• maximum stress in segment i is given by 6iF/(wih2i), a monomial
• the vertical deflection yi and slope vi of central axis at the right end of segment i are defined recursively as

for i = N,N −1,…,1, with vN+1 = yN+1 = 0 (E is Young’s modulus) vi and yi are posynomial functions of w, h
formulation as a GP
minimize subject to w1h1 + ··· + wNhN wmax−1 wi ≤ 1, wminwi−1 ≤ 1, i = 1,…,N h−max1 hi ≤ 1, hminh−i 1 ≤ 1, i = 1,…,N
Smax−1 wi−1hi ≤ 1, Sminwih−i 1 ≤ 1, i = 1,…,N
6iFσmax−1 wi−1h−i 2 ≤ 1, i = 1,…,N ymax−1 y1 ≤ 1
note
• we write wmin ≤ wi ≤ wmax and hmin ≤ hi ≤ hmax
wmin/wi ≤ 1, wi/wmax ≤ 1,
• we write Smin ≤ hi/wi ≤ Smax as hmin/hi ≤ 1, hi/hmax ≤ 1
Sminwi/hi ≤ 1, hi/(wiSmax) ≤ 1
Minimizing spectral radius of nonnegative matrix
Perron-Frobenius eigenvalue λpf(A)
• exists for (elementwise) positive A ∈ Rn×n
• a real, positive eigenvalue of A, equal to spectral radius maxi |λi(A)|
• determines asymptotic growth (decay) rate of
• alternative characterization: for some v ≻ 0}
minimizing spectral radius of matrix of posynomials
• minimize λpf(A(x)), where the elements A(x)ij are posynomials of x • equivalent geometric program:
minimize λ
subject to
variables λ, v, x
Generalized inequality constraints
convex problem with generalized inequality constraints
minimize f0(x) subject to
• f0 : Rn → R convex; fi : Rn → Rki Ki-convex w.r.t. proper cone Ki
• 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 cTx
subject to
extends linear programming ( ) to nonpolyhedral cones
Semidefinite program (SDP)
minimize cTx
subject to
with Fi, G ∈ Sk
• inequality constraint is called linear matrix inequality (LMI)
• includes problems with multiple LMI constraints: for example,

is equivalent to single LMI

LP and SOCP as SDP
LP and equivalent SDP
LP: minimize cTx SDP: minimize cTx subject to subject to diag
(note different interpretation of generalized inequality )
SOCP and equivalent SDP
SOCP: minimize fTx
subject to
SDP: minimize fTx
subject to
Eigenvalue minimization
minimize λmax(A(x))
where A(x) = A0 + x1A1 + ··· + xnAn (with given Ai ∈ Sk)
equivalent SDP
minimize t
subject to
• variables x ∈ Rn, t ∈ R
• follows from

Matrix norm minimization
minimize
where A(x) = A0 + x1A1 + ··· + xnAn (with given Ai ∈ Rp×q) equivalent SDP
minimize t subject to
• variables x ∈ Rn, t ∈ R
• constraint follows from

Vector optimization
general vector optimization problem
minimize (w.r.t. K) f0(x)
subject to fi(x) ≤ 0, hi(x) ≤ 0, i = 1,…,m
i = 1,…,p
vector objective f0 : Rn → Rq, minimized w.r.t. proper cone K ∈ Rq
convex vector optimization problem
minimize (w.r.t. K) f0(x)
subject to fi(x) ≤ 0, i = 1,…,m Ax = b
with f0 K-convex, f1, . . . , fm convex
Optimal and Pareto optimal points
set of achievable objective values
O = {f0(x) | x feasible}
• feasible x is optimal if f0(x) is a minimum value of O
• feasible x is Pareto optimal if f0(x) is a minimal value of O

x⋆ is optimal xpo is Pareto optimal
Multicriterion optimization
vector optimization problem with K = Rq+
f0(x) = (F1(x),…,Fq(x))
• q different objectives Fi; roughly speaking we want all Fi’s to be small
• feasible x⋆ is optimal if
y feasible
if there exists an optimal point, the objectives are noncompeting
• feasible xpo is Pareto optimal if
y feasible , f
if there are multiple Pareto optimal values, there is a trade-off between the objectives
Regularized least-squares
minimize (w.r.t. R

example for A ∈ R100×10; heavy line is formed by Pareto optimal points
Risk return trade-off in portfolio optimization
minimize (w.r.t. R subject to
• x ∈ Rn is investment portfolio; xi is fraction invested in asset i
• p ∈ Rn is vector of relative asset price changes; modeled as a random variable with mean p¯, covariance Σ
• p¯Tx = Er is expected return; xTΣx = varr is return variance example

0% 10% 20% 0% 10% 20% standard deviation of return standard deviation of return
Scalarization
to find Pareto optimal points: choose λ ≻K∗ 0 and solve scalar problem
minimize λTf0(x)
subject to fi(x) ≤ 0, hi(x) = 0, i = 1,…,m
i = 1,…,p
if x is optimal for scalar problem, then it is Pareto-optimal for vector optimization problem
for convex vector optimization problems, can find (almost) all Pareto optimal points by varying λ ≻K∗ 0
Scalarization for multicriterion problems
to find Pareto optimal points, minimize positive weighted sum
λTf0(x) = λ1F1(x) + ··· + λqFq(x)
examples
• regularized least-squares problem of page 1–43
take λ = (1,γ) with γ > 0 15
minimize kAx − bk22 + γkxk22
for fixed γ, a LS problem 5
• risk-return trade-off of page 1–44
minimize −p¯Tx + γxTΣx
subject to
for fixed γ > 0, a quadratic program