Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
Introduction to Optimization Problems andMethods
Jingchen [email protected]
December 10, 2009
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
Outline
1 Linear OptimizationProblemSimplex Method
2 Nonlinear OptimizationOne-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
3 Integer OptimizationCutting Plane Method
4 Dynamic OptimizationDiscrete Dynamic Programming
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
ProblemSimplex Method
General and Standard form of Linear Programming problem
General Form
min cT xs.t. aT
i x ≥ bi i ∈ M1
aTi x ≤ bi i ∈ M2
aTi x = bi i ∈ M3
xj ≥ 0 j ∈ N1
xj ≤ 0 j ∈ N2
Standard Form
min cT xs.t. Ax = b
x ≥ 0
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
ProblemSimplex Method
Simplex Method
Polyhedron Geometric Interpretation ofSimplex Method
If optimal solution exists, it canbe achieved on one of thevertices. The algorithm travelsalong edges of the polyhedron tovertices with higher functionvalue until it achieves optimalvalue.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
ProblemSimplex Method
Pros and Cons of Simplex Method
Pros
Remarkably efficient in practice, especially when the scale ofthe problem is small.
Cons
Worst-case complexity is exponential time.
For almost every pivoting rule, there is an exponentialworst-case complexity example.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
One-Dimensional Nonlinear Problem
Problem
minx∈[a,b] f (x)
Assumption
There is a unique local minimum in the interval considered.
f (x) may not be differentiable.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Bisection Method
5b
X5
5a
In each iteration, the lengthof the interval becomes halfof the previous one.
Calculate the value at twomore points in eachiteration.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Bisection Method
5b
X5
5a In each iteration, the lengthof the interval becomes halfof the previous one.
Calculate the value at twomore points in eachiteration.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Golden Section Method
b
a=
a
c=
b − c
c= 1.618
In each iteration, the lengthof the interval becomes0.618 of the previous one.
Only need to calculate onemore point in each iteration.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Golden Section Method
b
a=
a
c=
b − c
c= 1.618
In each iteration, the lengthof the interval becomes0.618 of the previous one.
Only need to calculate onemore point in each iteration.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Multi-Dimensional Unconstrained Problem
Problem
min f (x), x ∈ Rn
Assumption
f (x) is 2 times differentiable.
Questions:
How to choose direction?
How to choose step length?
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Steepest Descent Method
Choose di at xi as di = −∇f (xi ).
Since steepest descent direction is a local property, themethod is not effective.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Steepest Descent Method
Choose di at xi as di = −∇f (xi ).
Since steepest descent direction is a local property, themethod is not effective.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Conjugate Gradient Method (1)
Definition
Nonzero vectors u and v are conjugate with respect to positivedefinite matrix A if uTAv = 0
Property
If any two of nonzero vectors v1, v2, ...vn are conjugate in Rn, thenv1, v2, ..., vn form a base in Rn.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Conjugate Gradient Method (1)
Definition
Nonzero vectors u and v are conjugate with respect to positivedefinite matrix A if uTAv = 0
Property
If any two of nonzero vectors v1, v2, ...vn are conjugate in Rn, thenv1, v2, ..., vn form a base in Rn.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Conjugate Gradient Method (2)
Consider f (x) = 12xTAx − bT x , where A is positive definite.
rk = b − Axk is called residual, which is gradient descent.
Let
pk+1 = rk −∑i≤k
pTi Ark
pTi Api
pi .
xk+1 = xk + αk+1pk+1, where αk+1 =pTk+1rk
pTk+1Apk+1
.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Conjugate Gradient Method (2)
Consider f (x) = 12xTAx − bT x , where A is positive definite.
rk = b − Axk is called residual, which is gradient descent.
Let
pk+1 = rk −∑i≤k
pTi Ark
pTi Api
pi .
xk+1 = xk + αk+1pk+1, where αk+1 =pTk+1rk
pTk+1Apk+1
.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Conjugate Gradient Method (2)
Consider f (x) = 12xTAx − bT x , where A is positive definite.
rk = b − Axk is called residual, which is gradient descent.
Let
pk+1 = rk −∑i≤k
pTi Ark
pTi Api
pi .
xk+1 = xk + αk+1pk+1, where αk+1 =pTk+1rk
pTk+1Apk+1
.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Conjugate Gradient Method (2)
Consider f (x) = 12xTAx − bT x , where A is positive definite.
rk = b − Axk is called residual, which is gradient descent.
Let
pk+1 = rk −∑i≤k
pTi Ark
pTi Api
pi .
xk+1 = xk + αk+1pk+1, where αk+1 =pTk+1rk
pTk+1Apk+1
.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Conjugate Gradient Method (3)
Avoid repeating directions usedbefore.
Convergence in finite steps forquadratic objective functions.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Newton Method (1)
Consider f (x) = 12(x − xk)TA(x − xk)− bT (x − xk), where A
is positive definite.
The minimizer is xk+1 = xk + A−1b.
The idea of Newton Method is approximating a functionlocally at xk by its first three terms of Taylor expansion, andset the next iterate be the minimizer of the approximation.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Newton Method (1)
Consider f (x) = 12(x − xk)TA(x − xk)− bT (x − xk), where A
is positive definite.
The minimizer is xk+1 = xk + A−1b.
The idea of Newton Method is approximating a functionlocally at xk by its first three terms of Taylor expansion, andset the next iterate be the minimizer of the approximation.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Newton Method (2)Pros and Cons
Pros
Newton’s method can often converge remarkably quickly,especially if the iteration begins “sufficiently near” the optimalsolution.
A modification Quasi-Newton methods saves computation.
Cons
If f (x) is not convex function, the Newton’s method maysometimes diverge or converge to saddle point and localminimum.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Problem
Problem
min f (x) s.t. gi (x) ≤ 0, j = 1, 2, ...,m
Assumption
gi (x), j = 1, 2, ...,m are convex function. So feasible region isconvex.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Interior Penalty Function Method
Choose penalty function as1
gj(x), and let φ(x , rk) =
f (x)− rk
m∑j=1
1
gj(x).
Suppose we have an initialfeasible solution. Dounconstrained optimizationfor φ(x , rk) in each step.
let rk → 0 as k →∞. When rk is very small, φ approximatelyequal to the original constrained problem.
More often, penalty function log(gj(x)) is used.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
One-Dimensional OptimizationMulti-Dimensional Unconstrained OptimizationMulti-Dimensional Constrained Optimization
Interior Penalty Function Method
Choose penalty function as1
gj(x), and let φ(x , rk) =
f (x)− rk
m∑j=1
1
gj(x).
Suppose we have an initialfeasible solution. Dounconstrained optimizationfor φ(x , rk) in each step.
let rk → 0 as k →∞. When rk is very small, φ approximatelyequal to the original constrained problem.
More often, penalty function log(gj(x)) is used.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
Cutting Plane Method
Integer Programming-Cutting Plane Method
Blue lines are the boundary;
Red dots are feasible solutions;
Assume cost vector is verticallyupward;
Green line is the cutting plane.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
Discrete Dynamic Programming
A Discrete Dynamic Programming Example
There are T + 1 periods, t = 0, 1, ...,T .kt : capital in period t; ct : consumption in period t; u(ct)utility of consuming ct .
kt+1 = Akat − ct , A > 0, 0 < a < 1
Problem
maxT∑
t=0
λtu(ct), subject to kt+1 = Akat − ct ≥ 0
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
Discrete Dynamic Programming
Discrete Bellman Equation
Define value function VT+1(k) = 0;
Bellman Equation
Vt(kt) = max(u(ct) + λVt+1(kt+1))subject to kt+1 = Aka
t − ct ≥ 0 for t = 0, 1, ...,T
Bellman Equation can be solved by backward induction.
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
Discrete Dynamic Programming
A Continuous Dynamic Programming Example
Problem
min
{∫ T
0C [x(t), a(t)] + D[x(T )]
}where C (·) is the cost rate function and D(·) is the utility at thefinal state, x(t) is the system state vector, x(0) is given, and a(t)is control vector.The system is also subject to x(t) = F [x(t), u(t)].
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
Discrete Dynamic Programming
HJB Equation
Define value function V (x ,T ) = D(x);
V (x(t), t) = mina{C (x(t), a)dt + V (x(t + dt), t + dt)}
= mina{C (x(t), a)dt + V (x , t)dt +
∇V (x , t)xdt + o(dt2)}
HJB Equation
V (x , t) + mina{∇V (x , t)F (x , a) + C (x , a)} = 0
V (x ,T ) = D(x)
Jingchen Wu Introduction to Optimization Problems and Methods
Linear OptimizationNonlinear Optimization
Integer OptimizationDynamic Optimization
Discrete Dynamic Programming
Thank you very much for your attention!
Jingchen Wu Introduction to Optimization Problems and Methods