Conic Linear Programming and Applications · Conic Linear Optimization and Appl. MS&E 314 Lecture Note #01 15 Let V = UTUand z = Ux. Then the problem can be written as minimize α

Conic Linear Optimization and Appl. MS&E 314 Lecture Note #01 1

Conic Linear Programming and Applications

Yinyu Ye

Department of Management Science and Engineering

Stanford University

Stanford, CA 94305, U.S.A.

http://www.stanford.edu/˜yyye


Mathematical Programming (MP)

The class of mathematical programming problems considered in this course can

all be expressed in the form

(P) minimize f(x)

subject to x ∈ X

where X usually specified by constraints:

ci(x) = 0 i ∈ Eci(x) ≤ 0 i ∈ I.


Important Terms

• decision variable/activity, data/parameter

• objective/goal/target, coefficient vector/coefficient matrix

• constraint/limitation/requirement, satisfied/violated

• equality/inequality constraint, direction of inequality

• constraint function/the right-hand side

• nonnegativity constraint, integrality constraint

• feasible/infeasible solution

• global and local optimizers, global and local minimum values


A Special Class of MP Examples

minimize 2x1 + x2 + x3

subject to x1 + x2 + x3 = 1,

(x1;x2;x3) ≥ 0;


subject to x1 + x2 + x3 = 1,√x22 + x2

3 ≤ x1.

where

c = (2; 1; 1) and a1 = (1; 1; 1).



subject to x1 + x2 + x3 = 1,⎛⎝ x1 x2

x2 x3

⎞⎠ � 0,

where

c =

⎛⎝ 2 .5

.5 1

⎞⎠ and a1 =

⎛⎝ 1 .5

.5 1

⎞⎠ .


Conic Linear Programming/Optimization

(CLP ) minimize c • xsubject to ai • x = bi, i = 1, 2, ...,m, x ∈ C,

where C is a pointed and closed convex cone.

Linear Programming (LP): c, ai,x ∈ Rn and C = Rn+

Second-Order Cone Programming (SOCP): c, ai,x ∈ Rn and

C = Nn2 := {x : x1 ≥ ‖x−1‖2}

Semidefinite Programming (SDP): c, ai,x ∈ Sn and C = Sn+

Mixed Conic Linear Programming: C can be a product of closed convex cones

x = (x1;x2) where x1 ∈ C1 and x2 ∈ C2, and C1 and C2 are two different

cones.


Conic Linear Programming Characteristics

• linear objective function/constraints

• variables in a pointed and closed convex cone

• conic interior, boundary, extreme point (corner)

• every local optimizer is global, and they form a convex (optimizer) set

• most of them possess efficient algorithms in both practice and theory

(polynomial-time)


Conic Linear Optimization Model and Formulation

• Sort out data and parameters from the verbal description

• Define the set of decision variables

• Formulate the linear objective function of data and decision variables

• Set up linear equality and conic constraints


Example 1: Parimutuel Call Auction Mechanism I

Given m potential states that are mutually exclusive and exactly one of them will

be realized at the maturity.

An order is a bet on one or a combination of states, with a price limit (the

maximum price the participant is willing to pay for one unit of the order) and a

quantity limit (the maximum number of units or shares the participant is willing to

accept).

A contract on an order is a paper agreement so that on maturity it is worth a

notional $1 dollar if the order includes the winning state and worth $0 otherwise.

There are n orders submitted now.


Parimutuel Call Auction Mechanism II: order data

The ith order is given as (ai· ∈ Rm+ , πi ∈ R+, qi ∈ R+): ai· is the betting

indication row vector where each component is either 1 or 0

ai· = (ai1, ai2, ..., aim)

where 1 is winning state and 0 is non-winning state; πi is the price limit for one

unit of such a contract, and qi is the maximum number of contract units the better

like to buy.


Parimutuel Call Auction Mechanism III: order fills

Let xi be the number of units or shares awarded to the ith order. Then, the ith

bidder will pay the amount πi · xi and the total amount collected would be

πTx =∑

i πi · xi.

If the jth state is the winning state, then the auction organizer need to pay the

winning bidders (n∑

i=1

aijxi

)= aT·jx

where column vector

a·j = (a1j ; a2j ; ...; anj)

The question is, how to decide x ∈ Rn, that is, how to fill the orders.


Parimutuel Call Auction Mechanism IV: worst-case profit maximization

max πTx−maxj{aT·jx}s.t. x ≤ q,

x ≥ 0.

max πTx−max(ATx)

s.t. x ≤ q,

x ≥ 0.

This is NOT a linear program.


Parimutuel Call Auction Mechanism V: linear programming

However, the problem can be rewritten as

max πTx− y

s.t. ATx− e · y ≤ 0,

x ≤ q,

x ≥ 0,

where e is the vector of all ones. This is a linear program.

max πTx− y

s.t. ATx− e · y + s0 = 0,

x+ s = q,

(x, y, s0, s) ≥ 0,


Example 2: Portfolio Management

Let r denote the expected return vector and V denote the co-variance matrix of

an investment portfolio, and let x be the investment proportion vector. Then, one

management model is:

minimize xTV x

subject to rTx ≥ μ,

eTx = 1, x ≥ 0.

where e is the vector of all ones. This is a quadratic program.


Let V = UTU and z = Ux. Then the problem can be written as

minimize α

subject to Ux− z = 0,

rTx− s0 = μ,

eTx = 1,

(x; s0) ≥ 0, ‖z‖2 ≤ α,

which is a mixed linear and second-order cone program.


Robust Portfolio Management

In real applications, r and V may be estimated under various scenarios, say r i

and Vi for i = 1, ...,m.

minimize maxi xTVix

subject to mini rTi x ≥ μ,

eTx = 1, x ≥ 0.

minimize α

subject to rTi x ≥ μ, ∀ixTVix ≤ α, ∀ieTx = 1, x ≥ 0.

This can be further reduced to a mixed linear and second-order cone program.


Example 3: Graph Realization

Given a graph G = (V,E) and sets of non–negative weights, say

{dij : (i, j) ∈ E}, the goal is to compute a realization of G in the Euclidean

space Rd for a given low dimension d, i.e.

• to place the vertices of G in Rd such that

• the Euclidean distance between every pair of adjacent vertices (i, j) equals

(or bounded) by the prescribed weight dij ∈ E.


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 1: 50-node 2-D Graph Realization


Figure 2: A 3-D Tensegrity Graph Realization; picture provided by Anstreicher


Figure 3: Tensegrity Graph: A Needle Tower; picture provided by Anstreicher


Figure 4: Molecular Conformation: 1F39(1534 atoms) with 85% of distances below

6rA and 10% noise on upper and lower bounds


Graph Realization II: Distance Geometry Model

System of nonlinear equations for xi ∈ Rds:

‖xi − xj‖ = dij , ∀ (i, j) ∈ Nx, i < j,

‖ak − xj‖ = dkj , ∀ (k, j) ∈ Na,

where ak are possible points whose locations are known.


Graph Realization III: Nonlinear Optimization

Nonlinear least-squares:

min∑

i,j∈Nx(‖xi − xj‖2 − d2ij)

2 +∑

k,j∈Na(‖ak − xj‖2 − d2kj)

2

or

min∑

i,j∈Nx(‖xi − xj‖ − dij)

2 +∑

k,j∈Na(‖ak − xj‖ − dkj)

2

Either one is a non-convex optimization problem.


Graph Realization IV: Matrix Representation

Let X = [x1 x2 ... xn] be the 2× n matrix that needs to be determined. Then

‖xi − xj‖2 = (ei − ej)TXTX(ei − ej) and

‖ak − xj‖2 = (ak;−ej)T [I X ]T [I X ](ak;−ej),

where ei is the vector with 1 at the ith position and zero everywhere else.

(ei − ej)T (ei − ej) • Y = d2ij , ∀ i, j ∈ Nx, i < j,

(ak;−ej)T (ak;−ej) •

⎛⎝ I X

XT Y

⎞⎠ = d2kj , ∀ k, j ∈ Na,

Y −XTX = 0.

where Y denotes the Gram matrix XTX .


Graph Realization V: SDP Relaxation

Change

Y −XTX = 0

to

Y −XTX � 0.

This matrix inequality is equivalent to⎛⎝ I X

XT Y

⎞⎠ � 0.

This is the semidefinite matrix cone.


Graph Realization VI: SDP standard form

Z =

⎛⎝ I X

XT Y

⎞⎠ .

Find a symmetric matrix Z ∈ R(2+n)×(2+n) such that

Z1:2,1:2 = I

(0; ei − ej)(0; ei − ej)T • Z = d2ij , ∀ i, j ∈ Nx, i < j,

(ak;−ej)(ak;−ej)T • Z = d2kj , ∀ k, j ∈ Na,

Z � 0.

This is semidefinite programming with a null objective.

If the rank of Z is constrained to be d, then it is equivalent to the original graph

relaxation problem.


Example 4: The Kissing Problem

• Given a unit center sphere, the maximum number of unit spheres, in d

dimensions, can touch or kiss the center sphere?

• General Solutions does not exist.

• Delsarte Method uses linear programming to provide an upper bound on the

number of spheres.

• K(1)=2, K(2)=6, K(3)= 12, K(8) = 240, K(24) = 196650.

• K(4) = 24: proved using Delsarte Method by Oleg Musin only 3 years ago.

• For other dimensions, lower bounds have been provided by constructing a

lattice structure. There also exists a bound using the Riemann zeta function,

but is non-constructive.


The Kissing Problem as a Graph Realization

Can n balls all kiss the center ball at the same time?

• Does the following problem has a solution?

‖xi − xj‖2 = (ei − ej)TXTX(ei − ej) ≥ 4, ∀i �= j,

‖xi‖2 = eTi XTXei = 4, ∀i

where X = [x1 x2 ... xn] is the d× n matrix that needs to be determined

• Let Y = XTX . Then it can be formulated as a rank-constrained SDP

feasibility problem;

(ei − ej)(ei − ej) • Y ≥ 4, ∀i �= j,

eTi ei • Y = 4, ∀i,Y � 0,

rank(Y ) = d.


• The SDP feasibility relaxation is

(ei − ej)T (ei − ej) • Y ≥ 4, ∀i �= j,

eTi ei • Y = 4, ∀i,Y � 0.

• But the SDP solution may not provide a proper rank so that we construct a

nonzero SDP objective for the SDP relaxation

min C • Ys.t. (ei − ej)

T (ei − ej) • Y ≥ 4, ∀i �= j,

eTi ei • Y = 4, ∀i,Y � 0.

For 12 spheres, SDP method provides the following realization:


Figure 5: 12 Spheres in 3-D

Documents

Conic Linear Programming and Applications · Conic Linear Optimization and Appl. MS&E 314 Lecture Note #01 15 Let V = UTUand z = Ux. Then the problem can be written as minimize α