Constrained Optimization
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Transcript of Constrained Optimization
Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Constrained Optimization Slide 1 of 22
Constrained Optimization
Objective of Presentation: To introduce Lagrangean as a basic conceptual method used to optimize design in real situations
Essential Reality: In practical situations, the designers are constrained or limited by
physical realities design standards laws and regulations, etc.
Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Constrained Optimization Slide 2 of 22
Outline Unconstrained Optimization (Review)
Constrained Optimization – Lagrangeans Approach Lagrangeans as Equality constraints
Interpretation of Lagrangeans as “Shadow prices”
Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Constrained Optimization Slide 3 of 22
Unconstrained Optimization: Definitions
Optimization => Maximum of desired quantity, or => Minimum of undesired quantity
Objective Function = Formula to be optimized= Z(X)
Decision Variables = Variables about whichwe can make decisions= X = (X1….Xn)
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Unconstrained Optimization: Graph
B and D are maxima A, C and E are minima
A
B
C
D
E
F(X)
X
Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Constrained Optimization Slide 5 of 22
Unconstrained Optimization: Conditions By calculus: if F(X) continuous, analytic
Primary conditions for maxima and minima: F(X) / Xi = 0 i
( symbol means: “for all i”)
Secondary conditions:2F(X) / Xi
2 < 0 = > Max (B,D)2F(X) / Xi
2 > 0 = > Min (A,C,E)These define whether point of no change in Z is a
maximum or a minimum
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Unconstrained Optimization: Example Example: Housing insulation
Total Cost = Fuel cost + Insulation costx = Thickness of insulation
F(x) = K1 / x + K2x
Primary condition: F(x) / x = 0 = -K1 / x2 + K2
=> x* = {K1 / K2} 1/2
(starred quantities are optimal)
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Unconstrained Optimization: Graph of Solution to Example
Optimizing Cost Example
0100
200300400
500600
1 3 5 7 9 11
Inches of Insulation
Cost
FuelInsulationTotal
If: K1 = 500 ; K2 = 24 Then: X* = 4.56
Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Constrained Optimization Slide 8 of 22
Constrained Optimization: General “Constrained Optimization” involves the
optimization of a process subject to constraints
Constraints have two basic types Equality Constraints -- some factors have to
equal constraints Inequality Constraints -- some factors have to
be less less or greater than the constraints (these are “upper” and “lower” bounds)
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Constrained Optimization: General Approach
To solve situations of increasing complexity, (for example, those withequality, inequality constraints) …
Transform more difficult situation into one we know how to deal with
Note: this process introduces new variables!
Thus, transform “constrained” optimization to “unconstrained”
optimization
Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Constrained Optimization Slide 10 of 22
Equality Constraints: Example Example: Best use of budget Maximize: Output = Z(X) = a0x1
a1x2a2
Subject to (s.t.):Total costs = Budget = p1x1 + p2x2
Z(x) Z*
Budget X
Note: Z(X) / X 0 at optimum
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Lagrangean Method: Approach Transforms equality constraints into
unconstrained problem Start with:
Opt: F(x)s.t.: gj(x) = bj => gj(x) - bj = 0
Get to: L = F(x) - j j[gj(x) - bj]j = Lagrangean multipliers (lambdas sub j) -- are unknown quantities for which we must solve
Note: [gj(x) - bj] = 0 by definition, thus optimum for F(x) = optimum for L
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Lagrangean: Optimality Conditions
Since the new formulation is a single equation, we can use formulas for unconstrained optimization.
We set partial derivatives equal to zero for all unknowns, the X and the
Thus, to optimize L:L / xi = 0 I
L / j = 0 J
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Lagrangean: Example Formulation Problem:
Opt: F(x) = 6x1x2s.t.: g(x) = 3x1 + 4x2 = 18
Lagrangean: L = 6x1x2 - (3x1 + 4x2 - 18)
Optimality Conditions: L / x1 = 6x2 - 3 = 0
L / x2 = 6x1 - 4 = 0L / j = 3x1 + 4x2 -18 = 0
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Lagrangean: Graph for Example
Isoquants for F(X) = 20 and 40 and Constraint
-2.00-1.000.001.002.003.004.005.006.007.008.00
1 2 3 4 5
X (sub 1)
X (s
ub 2
)
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Lagrangean: Example Solution Solving as unconstrained problem: L / x1 = 6x2 - 3 = 0
L / x2 = 6x1 - 4 = 0L / i = 3x1 + 4x2 -18 = 0
so that: = 2x2 = 1.5x1 (first 2 equations) => x2 = 0.75x1
=> 3x1 + 3x1 - 18 = 0 (3rd equation) x1* = 18/6 = 3 x2* = 18/8 = 2.25 * = 4.5 F(x)* = 40.5
Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Constrained Optimization Slide 16 of 22
Lagrangean: Graph for Solution
Isoquants for F(X) = 20 and 40 and Constraint
-2.00-1.000.001.002.003.004.005.006.007.008.00
1 2 3 4 5
X (sub 1)
X (s
ub 2
)
Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Constrained Optimization Slide 17 of 22
Shadow Price Shadow Price = Rate of change of objective
function per unit change of constraint= F(x) / bj = j
It is extremely important for system design
It defines value of changing constraints, and indicates if worthwhile to change them
Should we buy more resources? Should we change environmental constraints?
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Lagrangean Multiplier is a Shadow Price
The Lagrangean multiplier is interpreted as the shadow price on constraint
SPj = F(x)*/ bj = L*/ bj = {F(x) - j j[gj(x) - bj] } / bj = j
Naturally, this is an instantaneous rate
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Lagrangean = Shadow Prices Example
Let’s see how this works in example, by changing constraint by 0.1 units:
Opt: F(x) = 6x1x2s.t.: g(x) = 3x1 + 4x2 = 18.1
The optimum values of the variables are x1* = (18.1)/6 x2* = (18.1)/8 * = 4.5
Thus F(x)* = 6(18.1/6)(18.1/8) = 40.95
F(x) = 40.95 - 40.5 = 0.45 = * (0.1)
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Generalization In general constraints are “inequalities”:
Upper bounds : gj(x) < bj Lower bounds: gj(x) > bj
At optimum, some constraints will limit solution (they are “binding”) others not
Example: airline bags: weight < 40kg ; sum of dimensions < 2.5m. Your bag might be limited by weight, not by size.
Shadow prices > 0 for all “binding” constraints= 0 for all others
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Design Implications Expanding range of design variables (x),
increases freedom to improve design, thus adds value
This is called “relaxing” the constraints Increasing upper bounds Decreasing lower bounds
As any constraint relaxed, it may no longer be “binding” , and others can become so
SHADOW PRICES DEPEND ON OTHER CONSTRAINTS, “PROBLEM DEPENDENT”
Engineering Systems Analysis for Design Richard de Neufville © Massachusetts Institute of Technology Constrained Optimization Slide 22 of 22
Take-aways
Relaxing design constraints adds value (in terms of better performance, F(x) )
This value is the “shadow price” of that constraint
Knowing this can be very important for designers, shows way to improve quickly
NOTE: Value to design has no direct connection to cost of constraint, not a “price in ordinary terms