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### Boundary oracle and its use in convex optimization

Boris Polyak

Institute for Control Science

Moscow Russia

Luminy, April 2007

Contents

- Convex optimization problem
- Boundary oracle
- Examples
- Center of gravity and Radon theorem
- Algorithm 1: Hit-and-Run
- Algorithm 2: Stochastic approximation
- Algorithm 3: Monte-Carlo
- Implementation
- Simulation
- Future directions

Convex optimization problem

x*

max (c,x)

x K

K

d

K is a convex closed

bounded set in Rn

Main problem: random vs deterministic algorithms

Boundaryoracle

L

B

Given x є K, d є Rn

T=max{t: x +td K}

B=x+Td = B(x,d,K)

L is support vector to K

in B: (L,B)(L,y), y K

{B,L} – boundary oracle

K

d

x

Examples of K

- Boundary oracle is available for many sets K:
- Linear inequalities (LI) Axb
- Quadratic inequalities (QI) (Aix,x)+(bi,x)gi,i=1,…,m,
- Linear matrix inequalities (LMI) A0+xiAi 0
- Solution: A= A0+xiAi, C= –, diAi , i=eig(A,C),,
- T = k = min {i: i0}, B = x+Td, ek – eigenvector corresponding
- to k , L = ((A1ek,ek), … , (Anek,ek))
- Conic quadratic inequalities (CQI) Aix+bi (ci, x)+ gi
- Nonnegative polynomials x(s)=x0+x1s+…xnsn 0 s
- Solution: Let si be real roots of the polynomial x(s)d(s)+x(s)d(s).
- Then T=min{- x(si)/d(si): d(si)<0}, b(s)=x(s)+Td(s), s*- the single
- real root of b(s), L=(1, s*, …, s*n).

Robust versions of K

- Boundary oracle can be also constructed for robust
- versions of the above sets. Examples:
- Robust linear inequalities (RLI) K={x: (A+A)x
- b+b,, A A, b b}

2. Robust linear matrix inequalities (RLMI) K=

{x: A0+0+xi(Ai+i) 0, i i}

Center of gravity and Radon theorem

Let g be the center of gravity of a convex set K in Rn.

Radon theorem (1916)

If f+=maxxK (c,x), f-=minxK (c,x), fg=(c,g)

then 1/n(f+- fg)/(fg- f-) n

Worst case – a pyramid

f+

g

n/(n+1)

f-

1/(n+1)

fg

Idea of algorithms

max xK (c,x)

Find approximate center of gravity g for K using random

search and boundary oracle. Then construct

Knew = K{x: (c,x)(c,g)}

For Knew boundary oracle is also available and we can proceed.

If fk is the value of (c,x) obtained at k-th iteration and f* is the

optimal value, we can expect geometric rate of convergence:

fk-f*(f0-f*)qk, q=n/(n+1)

Values of L can be exploited to calculate upper bound for f*.

Comparison with center of gravity and ellipsoids methods

MinxK f(x) f, K –convex, g – center of gravity for K

Knew = K{x: (f(g),x-g)0}

Newman (1965), Levin (1965)

Another validation : Vol(K)/Vol(Knew), Grunbaum theorem.

How to find g?

Ellipsoids method – K and Knew are placed into ellipsoids.

Lower rate of convergence.

Algorithm 1- Hit-and-Run

Hit-and-Run (Smith 1984, Lovasz 1999) is an algorithm

for generating uniformly distributed points in a convex set

K. It exploits boundary oracle. The arithmetic mean of the points can be taken as an approximation for g.

This method was used for convex minimization by

Bertsimas and Vempala (2004).

The simplest and well working version was: generate points

xk recursively:

xk+1= rkB(xk, dk)+ (1-rk)B(xk, - dk), rk=rand

where dk is random direction uniformly distributed on the

unit sphere. Then after N samples the estimate for g is

gN= 1/Nxk

Algorithm 2 – stochastic approximation

xk+1=xk+gkTkdk, gk1/k, Tk=T(xk,dk)

dk is random uniformly distributed on the unit sphere

Then xkg.

There are many versions of the algorithm.

Algorithm 3 – Monte-Carlo

gk is the center of gravity of the

element of volume dVk,

gk=a+(n/(n+1))Tkdk,

dVk=gTkndSk,

g=gkdVk/ dVk

Thus the estimate for g after N

samples generated is

Bk=Tkdk

gk

dk

a

a

gN= a+nTkn+1dk/(n+1) Tkn

General scheme of the method

- Choose aK, e>0.
- Starting from a, generate approximation gN and oracles
- Bk, Lk, k=1,…,N. Calculate f–=max (c,Bk), f+=max {(c,x):
- (Lk, x-Bk)_0}.
- If f+- f–<e, stop. Else
- KK{x: (c,x-gN) 0 }, agN.

Implementation

- Finding gN is the best if K is ball-like and a is close to its center.
- Hence we can reshape K using transformation
- W=1/N (Bk-a) (Bk-a) T
- Accelerating step. We can use relaxation: if Gi, Gi+1 are two
- successive estimates for center of gravity, a= Gi+1 +l(Gi+1 – Gi),
- l>0 is better approximation for a.
- Choosing N adaptively.
- Final polishing.
- +many other tricks to accelerate convergence.

Simulation results – SDP

SDP: max(c,x) s.t. LMI constraints A(x)<0,

A(x)=A0+x1A1+…+xnAn Ai – m x m matrices

Results comparable with standard SDP solvers (YALMIP).

n~40, m~40, N~1000, k~15 ε~10-8

n~300, m~10, N~2000, k~15 ε~10-7

n~10, m~100, N~200, k~15 ε~10-9

Simulation for robust SDP (low-dimensional)

Rigorous validation (complexity estimation)

- Random algorithms for LP
- Convex problems with the lack of interior-point methods (nonnegative polynomials?)
- In general: can random algorithms be competitors with deterministic ones in convex optimization?

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