Prof. (Dr.) Rajib Kumar Bhattacharjya
Transcript of Prof. (Dr.) Rajib Kumar Bhattacharjya
Multivariable problem with equality and inequality constraints
Prof. (Dr.) Rajib Kumar Bhattacharjya
Professor, Department of Civil EngineeringIndian Institute of Technology Guwahati, India
Room No. 005, M BlockEmail: [email protected], Ph. No 2428
Rajib Bhattacharjya, IITG CE 602: Optimization Method
General formulation
๐ ๐Min/Max
Subject to ๐๐ ๐ = 0 ๐ = 1,2,3, โฆ ,๐
Where ๐ = ๐ฅ1, ๐ฅ2, ๐ฅ3, โฆ , ๐ฅ๐๐
This is the minimum point of the function
g ๐ = 0
Now this is not the minimum point of the constrained function
This is the new minimum point
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Consider a two variable problem
๐ ๐ฅ1, ๐ฅ2Min/Max
Subject to ๐ ๐ฅ1, ๐ฅ2 = 0
g ๐ฅ1, ๐ฅ2 = 0
๐ฅ1, ๐ฅ2
๐๐ =๐๐
๐๐ฅ1๐๐ฅ1 +
๐๐
๐๐ฅ2๐๐ฅ2 = 0
Take total derivative of the function at ๐ฅ1, ๐ฅ2
If ๐ฅ1, ๐ฅ2 is the solution of the constrained problem, then
๐ ๐ฅ1, ๐ฅ2 = 0
Now any variation ๐๐ฅ1 and ๐๐ฅ2 is admissible only when
๐ ๐ฅ1 + ๐๐ฅ1, ๐ฅ2 + ๐๐ฅ2 = 0
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Consider a two variable problem
๐ ๐ฅ1, ๐ฅ2Min/Max
Subject to ๐ ๐ฅ1, ๐ฅ2 = 0
g ๐ฅ1, ๐ฅ2 = 0
๐ฅ1, ๐ฅ2
๐ ๐ฅ1 + ๐๐ฅ1, ๐ฅ2 + ๐๐ฅ2 = 0
This can be expanded as
๐ ๐ฅ1 + ๐๐ฅ1, ๐ฅ2 + ๐๐ฅ2 = ๐ ๐ฅ1, ๐ฅ2 +๐๐ ๐ฅ1,๐ฅ2
๐๐ฅ1๐๐ฅ1 +
๐๐ ๐ฅ1,๐ฅ2
๐๐ฅ2๐๐ฅ2 = 0
0
๐๐ =๐๐
๐๐ฅ1๐๐ฅ1 +
๐๐
๐๐ฅ2๐๐ฅ2 = 0
๐๐ฅ2 = โ
๐๐๐๐ฅ1๐๐๐๐ฅ2
๐๐ฅ1
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Consider a two variable problem
๐ ๐ฅ1, ๐ฅ2Min/Max
Subject to ๐ ๐ฅ1, ๐ฅ2 = 0
g ๐ฅ1, ๐ฅ2 = 0
๐ฅ1, ๐ฅ2
๐๐ฅ2 = โ
๐๐๐๐ฅ1๐๐๐๐ฅ2
๐๐ฅ1
Putting in ๐๐ =๐๐
๐๐ฅ1๐๐ฅ1 +
๐๐
๐๐ฅ2๐๐ฅ2 = 0
๐๐ =๐๐
๐๐ฅ1๐๐ฅ1 โ
๐๐
๐๐ฅ2
๐๐๐๐ฅ1๐๐๐๐ฅ2
๐๐ฅ1 = 0
๐๐
๐๐ฅ1
๐๐
๐๐ฅ2โ๐๐
๐๐ฅ2
๐๐
๐๐ฅ1๐๐ฅ1 = 0
This is the necessary condition for optimality for optimization problem with equality constraints
๐๐
๐๐ฅ1
๐๐
๐๐ฅ2โ๐๐
๐๐ฅ2
๐๐
๐๐ฅ1= 0
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Lagrange Multipliers
๐ ๐ฅ1, ๐ฅ2Min/Max
Subject to ๐ ๐ฅ1, ๐ฅ2 = 0
We have already obtained the condition that
By defining
๐๐
๐๐ฅ1โ๐๐
๐๐ฅ2
๐๐
๐๐ฅ1๐๐
๐๐ฅ2
= 0๐๐
๐๐ฅ1โ
๐๐
๐๐ฅ2๐๐
๐๐ฅ2
๐๐
๐๐ฅ1= 0
๐ = โ
๐๐๐๐ฅ2๐๐๐๐ฅ2
We have๐๐
๐๐ฅ1+ ๐๐๐
๐๐ฅ1= 0
We can also write
๐๐
๐๐ฅ2+ ๐๐๐
๐๐ฅ2= 0
Necessary conditions for optimality
๐ ๐ฅ1, ๐ฅ2 = 0Also put
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Lagrange Multipliers
๐ฟ ๐ฅ1, ๐ฅ2, ๐ = ๐ ๐ฅ1, ๐ฅ2 + ๐๐ ๐ฅ1, ๐ฅ2
Let us define
By applying necessary condition of optimality, we can obtain
๐๐ฟ
๐๐ฅ1=๐๐
๐๐ฅ1+ ๐๐๐
๐๐ฅ1= 0
๐๐ฟ
๐๐ฅ2=๐๐
๐๐ฅ2+ ๐๐๐
๐๐ฅ2= 0
๐๐ฟ
๐๐= ๐ ๐ฅ1, ๐ฅ2 = 0
Necessary conditions for optimality
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Lagrange Multipliers
๐ฟ ๐ฅ1, ๐ฅ2, ๐ = ๐ ๐ฅ1, ๐ฅ2 + ๐๐ ๐ฅ1, ๐ฅ2
Sufficient condition for optimality of the Lagrange function can be written as
๐ป =
๐2๐ฟ
๐๐ฅ1๐๐ฅ1
๐2๐ฟ
๐๐ฅ1๐๐ฅ2
๐2๐ฟ
๐๐ฅ1๐๐
๐2๐ฟ
๐๐ฅ2๐๐ฅ1
๐2๐ฟ
๐๐ฅ2๐๐ฅ2
๐2๐ฟ
๐๐ฅ2๐๐
๐2๐ฟ
๐๐๐๐ฅ1
๐2๐ฟ
๐๐๐๐ฅ2
๐2๐ฟ
๐๐๐๐
๐ป =
๐2๐ฟ
๐๐ฅ1๐๐ฅ1
๐2๐ฟ
๐๐ฅ1๐๐ฅ2
๐๐
๐๐ฅ1๐2๐ฟ
๐๐ฅ2๐๐ฅ1
๐2๐ฟ
๐๐ฅ2๐๐ฅ2
๐๐
๐๐ฅ2๐๐
๐๐ฅ1
๐๐
๐๐ฅ20
If ๐ป is positive definite , the optimal solution is a minimum point
If ๐ป is negative definite , the optimal solution is a maximum point
Else it is neither minima nor maxima
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Lagrange Multipliers
Necessary conditions for general problem
๐ ๐Min/Max
Subject to ๐๐ ๐ = 0 ๐ = 1,2,3, โฆ ,๐
Where ๐ = ๐ฅ1, ๐ฅ2, ๐ฅ3, โฆ , ๐ฅ๐๐
๐ฟ ๐ฅ1, ๐ฅ2, โฆ , ๐ฅ๐, ๐1, ๐2, ๐3โฆ , ๐๐ = ๐ ๐ + ๐1๐1 ๐ + ๐2๐2 ๐ ,โฆ , ๐๐๐๐ ๐
Necessary conditions
๐๐ฟ
๐๐ฅ๐=๐๐
๐๐ฅ๐+
๐=1
๐
๐๐๐๐๐
๐๐ฅ๐= 0
๐๐ฟ
๐๐๐= ๐๐ ๐ = 0
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Lagrange Multipliers
Sufficient condition for general problem
๐ป =
๐ฟ11 ๐ฟ12 ๐ฟ13๐ฟ21 ๐ฟ22 ๐ฟ23โฎ โฎ โฎ
โฆ ๐ฟ1๐ ๐11โฆ ๐ฟ2๐ ๐12โฎ โฎ โฎ
๐21 โฆ ๐๐1๐22 โฆ ๐๐2โฎ โฎ โฎ
๐ฟ๐1 ๐ฟ๐2 ๐ฟ๐3๐11 ๐12 ๐13๐21 ๐22 ๐23
โฆ ๐ฟ๐๐ ๐1๐โฆ ๐1๐ 0โฎ ๐2๐ โฎ
๐1๐ โฆ ๐๐๐0 โฆ 0โฎ โฎ โฎ
๐31 ๐32 ๐33โฎ โฎ โฎ๐๐1 ๐๐2 ๐๐3
โฆ ๐3๐ 0โฆ โฎ โฎโฆ ๐2๐ 0
โฆ โฆ โฆโฆ โฆ โฆโฆ โฆ 0
The hessian matrix is Where,
๐ฟ๐๐=๐2๐ฟ
๐๐ฅ๐๐๐ฅ๐
๐๐๐=๐๐๐
๐๐ฅ๐
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Lagrange Multipliers
๐ ๐Min/Max
Subject to ๐ ๐ = ๐ Or, ๐ โ ๐ ๐ = 0
Applying necessary conditions
๐๐
๐๐ฅ๐โ ๐๐๐
๐๐ฅ๐= 0
Where ๐ = ๐ฅ1, ๐ฅ2, ๐ฅ3, โฆ , ๐ฅ๐๐
Where, ๐ = 1,2,3, โฆ , ๐
๐ โ ๐ = 0
Further ๐๐ โ ๐๐ = 0
๐๐ = ๐๐ =
๐=1
๐๐๐
๐๐ฅ๐๐๐ฅ๐
๐๐
๐๐ฅ๐=
๐๐๐๐ฅ๐๐
๐๐ =
๐=1
๐1
๐
๐๐
๐๐ฅ๐๐๐ฅ๐
๐๐ =๐๐
๐
๐ =๐๐
๐๐
๐๐ = ๐๐๐
There may be three conditions
๐ โ > 0๐ โ < 0๐ โ = 0
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Multivariable problem with inequality constraints
๐ ๐Minimize
Subject to ๐๐ ๐ โค 0 ๐ = 1,2,3, โฆ ,๐
Where ๐ = ๐ฅ1, ๐ฅ2, ๐ฅ3, โฆ , ๐ฅ๐๐
We can write ๐๐ ๐ + ๐ฆ๐2 = 0
Thus the problem can be written as
๐ ๐Minimize
Subject to ๐บ๐ ๐, ๐ = ๐๐ ๐ + ๐ฆ๐2 = 0 ๐ = 1,2,3, โฆ ,๐
Where Y= ๐ฆ1, ๐ฆ2, ๐ฆ3, โฆ , ๐ฆ๐๐
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Multivariable problem with inequality constraints
๐ ๐Minimize
Subject to ๐บ๐ ๐, ๐ = ๐๐ ๐ + ๐ฆ๐2 = 0 ๐ = 1,2,3, โฆ ,๐
Where ๐ = ๐ฅ1, ๐ฅ2, ๐ฅ3, โฆ , ๐ฅ๐๐
The Lagrange function can be written as
๐ฟ ๐, ๐, ๐ = ๐ ๐ +
๐=1
๐
๐๐๐บ๐ ๐, ๐
The necessary conditions of optimality can be written as
๐๐ฟ ๐, ๐, ๐
๐๐ฅ๐=๐๐ ๐
๐๐ฅ๐+
๐=1
๐
๐๐๐๐๐ ๐
๐๐ฅ๐= 0
๐๐ฟ ๐, ๐, ๐
๐๐๐= ๐บ๐ ๐, ๐ = ๐๐ ๐ + ๐ฆ๐
2 = 0
๐ = 1,2,3, โฆ , ๐
๐ = 1,2,3, โฆ ,๐
๐๐ฟ ๐,๐,๐
๐๐ฆ๐=2๐๐๐ฆ๐=0 ๐ = 1,2,3, โฆ ,๐
Rajib Bhattacharjya, IITG CE 602: Optimization Method
๐๐ฟ ๐,๐,๐
๐๐ฆ๐= 2๐๐๐ฆ๐=0
Multivariable problem with inequality constraints
From equation
Either ๐๐= 0 Or, ๐ฆ๐= 0
If ๐๐= 0, the constraint is not active, hence can be ignored
If ๐ฆ๐= 0, the constraint is active, hence have to consider
Now, consider all the active constraints, Say set ๐ฝ1 is the active constraintsAnd set ๐ฝ2 is the active constraints
The optimality condition can be written as๐๐ ๐
๐๐ฅ๐+
๐โ๐ฝ1
๐๐๐๐๐ ๐
๐๐ฅ๐= 0 ๐ = 1,2,3, โฆ , ๐
๐๐ ๐ = 0
๐๐ ๐ + ๐ฆ๐2 = 0
๐ โ ๐ฝ1
๐ โ ๐ฝ2
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Multivariable problem with inequality constraints
โ๐๐
๐๐ฅ๐= ๐1๐๐1๐๐ฅ๐+ ๐2๐๐2๐๐ฅ๐+ ๐3๐๐3๐๐ฅ๐+โฏ+ ๐๐
๐๐๐๐๐ฅ๐
๐ = 1,2,3, โฆ , ๐
โ๐ป๐ = ๐1๐ป๐1 + ๐2๐ป๐2 + ๐3๐ป๐3 +โฏ+ ๐๐๐ป๐๐
๐ป๐ =
๐๐๐๐ฅ1
๐๐๐๐ฅ2โฎ
๐๐๐๐ฅ๐
๐ป๐๐ =
๐๐๐๐๐ฅ1
๐๐๐๐๐ฅ2โฎ
๐๐๐๐๐ฅ๐
This indicates that negative of the gradient of the objective function can be expressed as a linear combination of the gradients of the active constraints at optimal point.
โ๐ป๐ = ๐1๐ป๐1 + ๐2๐ป๐2
Let ๐ be a feasible direction, then we can write
โ๐๐๐ป๐ = ๐1๐๐๐ป๐1 + ๐2๐
๐๐ป๐2
Since ๐ is a feasible direction ๐๐๐ป๐1 < 0 and ๐๐๐ป๐2 < 0
If ๐1, ๐2 > 0Then the term ๐๐๐ป๐ is +ve
This indicates that ๐ is a direction of increasing function value
Thus we can conclude that if ๐1, ๐2 > 0, we will not get any better solution than the current solution
Rajib Bhattacharjya, IITG CE 602: Optimization Method
๐1 ๐ = 0
๐2 ๐ = 0
๐1 ๐ < 0
๐2 ๐ < 0
๐1 ๐ > 0
๐2 ๐ > 0
๐ป๐1
๐ป๐2
๐
Infeasible region
Infeasible region
Feasible region
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Multivariable problem with inequality constraints
The necessary conditions to be satisfied at constrained minimum points ๐โ are
๐๐ ๐
๐๐ฅ๐+
๐โ๐ฝ1
๐๐๐๐๐ ๐
๐๐ฅ๐= 0 ๐ = 1,2,3, โฆ , ๐
๐๐ โฅ 0 ๐ โ ๐ฝ1
These conditions are called Kuhn-Tucker conditions, the necessary conditions to be satisfied at a relative minimum of ๐ ๐ .
These conditions are in general not sufficient to ensure a relative minimum, However, in case of a convex problem, these conditions are the necessary and sufficient conditions for global minimum.
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Multivariable problem with inequality constraints
If the set of active constraints are not known, the Kuhn-Tucker conditions can be stated as
๐๐ ๐
๐๐ฅ๐+
๐=1
๐
๐๐๐๐๐ ๐
๐๐ฅ๐= 0 ๐ = 1,2,3, โฆ , ๐
๐๐๐๐ = 0
๐๐ โฅ 0
๐๐ โค 0 ๐ = 1,2,3, โฆ ,๐
Rajib Bhattacharjya, IITG CE 602: Optimization Method
Multivariable problem with equality and inequality constraints
For the problem
๐๐ ๐
๐๐ฅ๐+
๐=1
๐
๐๐๐๐๐ ๐
๐๐ฅ๐+
๐=1
๐
๐ฝ๐๐โ๐ ๐
๐๐ฅ๐= 0 ๐ = 1,2,3, โฆ , ๐
๐๐๐๐ = 0
๐๐ โฅ 0
๐๐ โค 0
๐ = 1,2,3, โฆ ,๐
๐ ๐Minimize
Subject to ๐๐ ๐ = 0 ๐ = 1,2,3, โฆ ,๐
Where ๐ = ๐ฅ1, ๐ฅ2, ๐ฅ3, โฆ , ๐ฅ๐๐
๐๐ ๐ = 0 ๐ = 1,2,3, โฆ , ๐
The Kuhn-Tucker conditions can be written as
๐ = 1,2,3, โฆ ,๐
โ๐ = 0
๐ = 1,2,3, โฆ ,๐
๐ = 1,2,3, โฆ , ๐
Rajib Bhattacharjya, IITG CE 602: Optimization Method