Linear programming, quadratic programming, sequential quadratic

Report
CS B553: ALGORITHMS FOR
OPTIMIZATION AND LEARNING
Linear programming, quadratic programming,
sequential quadratic programming
KEY IDEAS

Linear programming
Simplex method
 Mixed-integer linear programming

Quadratic programming
 Applications

RADIOSURGERY
3
CyberKnife (Accuray)
Normal tissue
Tumor
Tumor
Radiologically
sensitive tissue
Tumor
Tumor
Tumor
Tumor
OPTIMIZATION FORMULATION
Dose cells (xi,yj,zk) in a voxel grid
 Cell class: normal, tumor, or sensitive
 Beam “images”: B1,…,Bn describing dose absorbed
at each cell with maximum power
 Optimization variables: beam powers x1,…,xn
 Constraints:

Normal cells: Dijk  Dnormal
 Sensitive cells: Dijk  Dsensitive
 Tumor cells: Dmin  Dijk  Dmax
 0  xb  1





=1  

Dose calculation: Dijk =

Objective: minimize total dose
 
LINEAR PROGRAM

General form
min fTx+g
s.t.
Axb
Cx=d
A convex polytope
A slice through the polytope
THREE CASES
Infeasible
Feasible, bounded
Feasible, unbounded
f
f
?
x*
x*
f
SIMPLEX ALGORITHM (DANTZIG)
Start from a vertex of the feasible polytope
 “Walk” along polytope edges while decreasing
objective on each step
 Stop when the edge is unbounded or no
improvement can be made


Implementation details:
How to pick an edge (exiting and entering)
 Solving for vertices in large systems
 Degeneracy: no progress made due to objective vector
being perpendicular to edges

COMPUTATIONAL COMPLEXITY
Worst case exponential
 Average case polynomial (perturbed analysis)
 In practice, usually tractable


Commercial software (e.g., CPLEX) can handle
millions of variables/constraints!
SOFT CONSTRAINTS
Penalty
Normal
Dose
Sensitive
Tumor
SOFT CONSTRAINTS
Auxiliary variable zijk: penalty at each cell
zijk
zijk  0
Dijk
zijk  c(Dijk – Dnormal)
Dose
SOFT CONSTRAINTS
Auxiliary variable zijk: penalty at each cell
fijk
zijk  0
zijk
zijk  c(Dijk – Dnormal)
Dose
Introduce term in objective to
minimize zijk
MINIMIZING AN ABSOLUTE VALUE

Absolute value
Objective
minx |x1|
s.t.
Ax  b
Cx = d
x1
Constraints
minv,x v
Ax  b
Cx = d
x1  v
-x1  v
s.t.
x1
MINIMIZING AN L-1 OR L-INF NORM


L1 norm
Fx*
minx ||Fx-g||1
s.t.
Ax  b
Cx = d
g
Feasible polytope,
projected thru F
Feasible polytope,
projected thru F
L norm
minx ||Fx-g||
s.t.
Ax  b
Cx = d
g
Fx*
MINIMIZING AN L-1 OR L-INF NORM

L1 norm
minx ||Fx-g||1
s.t.
Ax  b
Cx = d
mine,x 1Te
s.t.
Fx + Ie  g
Fx - Ie  g
Ax  b
Cx = d
Fx*
e
g
Feasible polytope,
projected thru F
MINIMIZING AN L-2 NORM

L2 norm
minx ||Fx-g||2
s.t.
Ax  b
Cx = d
Fx*
g
Not a linear program!
Feasible polytope,
projected thru F
QUADRATIC PROGRAMMING

General form
min ½ xTHx + gTx + h
s.t.
Axb
Cx=d
Objective: quadratic form
Constraints: linear
QUADRATIC PROGRAMS

H positive definite
Feasible polytope
H-1 g
QUADRATIC PROGRAMS

H positive definite
Optimum can lie off of a vertex!
H-1 g
QUADRATIC PROGRAMS

H negative definite
Feasible polytope
QUADRATIC PROGRAMS

H positive semidefinite
Feasible polytope
SIMPLEX ALGORITHM FOR QPS
Start from a vertex of the feasible polytope
 “Walk” along polytope facets while decreasing
objective on each step
 Stop when the facet is unbounded or no
improvement can be made


Facet: defined by mn constraints
m=n: vertex
 m=n-1: line
 m=1: hyperplane
 m=0: entire space

ACTIVE SET METHOD
Active inequalities S=(i1,…,im)
 Constraints ai1Tx = bi1, … aimTx = bim


Written as ASx – bS = 0

Objective ½ xTHx + gTx + f

Lagrange multipliers  = (1,…,m)
Hx + g + AST  = 0
 Asx - bS = 0



Solve linear system:

If x violates a different
constraint not in S, add it
−
 
= 

0

If k<0 , then drop ik from S
PROPERTIES OF ACTIVE SET METHODS FOR
QPS
Inherits properties of simplex algorithm
 Worst case: exponential number of facets
 Positive definite H: polynomial time in typical
case
 Indefinite or negative definite H: can be
exponential time!


NP complete problems
APPLYING QPS TO NONLINEAR PROGRAMS


Recall: we could convert an equality constrained
optimization to an unconstrained one, and use Newton’s
method
Each Newton step:
 Fits a quadratic form to the objective
 Fits hyperplanes to each equality
 Solves for a search direction (x,) using the linear
equality-constrained optimization
How about inequalities?
SEQUENTIAL QUADRATIC PROGRAMMING

Idea: fit half-space constraints to each inequality

g(x)  0 becomes g(xt) + g(xt)T(x-xt)  0
xt
g(x)  0
g(xt) + g(xt)T(x-xt)  0
SEQUENTIAL QUADRATIC PROGRAMMING

Given nonlinear minimization


minx f(x)
s.t.
gi(x)  0,
hj(x) = 0,
for i=1,…,m
for j=1,…,p
At each step xt, solve QP

minx ½xTx2L(xt,t,t)x + xL(xt,t,t)Tx
s.t.
gi(xt) + gi(xt)Tx  0
for i=1,…,m
hj(xt) + hj(xt)Tx = 0
for j=1,…,p
To derive the search direction x
 Directions  and  are taken from QP
multipliers

ILLUSTRATION
xt
x
g(x)  0
g(xt) + g(xt)T(x-xt)  0
ILLUSTRATION
x
xt+1
g(x)  0
g(xt+1) + g(xt+1)T(x-xt+1)  0
ILLUSTRATION
xt+2
x
g(x)  0
g(xt+2) + g(xt+2)T(x-xt+2)  0
SQP PROPERTIES
Equivalent to Newton’s method without
constraints
 Equivalent to Lagrange root finding with only
equality constraints
 Subtle implementation details:

Does the endpoint need to be strictly feasible, or just
up to a tolerance?
 How to perform a line search in the presence of
inequalities?


Implementation available in Matlab. FORTRAN
packages too =(

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