Constrained Optimization. Unconstrained Optimization (1)

Transcription

1 Constrained Optimization Unconstrained Optimization (Review) Constrained Optimization Approach Equality constraints * Lagrangeans * Shadow prices Inequality constraints * Kuhn-Tucker conditions * Complementary slackness Massachusetts Institute of Technology Constrained Optimization Slide 1 of 23 Unconstrained Optimization (1) Definitions: Optimization = Maximum of desired quantity = Minimum of undesired quantity Objective Function = Expression to be optimized = Z(X) Decision Variables = Variables about which we can make decisions = X = (X 1.X n ) Massachusetts Institute of Technology Constrained Optimization Slide 2 of 23 Page 1

2 Unconstrained Optimization (2) F(X) B D A By calculus: C E X If F(X) continuous, analytic: Condition for maxima and minima F(X) / X i = 0 i Massachusetts Institute of Technology Constrained Optimization Slide 3 of 23 Unconstrained Optimization (3) Secondary conditions: 2 F(X) / X 2 i < 0 = > Max (B,D) 2 F(X) / X 2 i > 0 = > Min (A,C,E) These define whether point of no change in Z is a maximum or a minimum Massachusetts Institute of Technology Constrained Optimization Slide 4 of 23 Page 2

3 Unconstrained Optimization (4) Example: Housing insulation F(x) = K 1 / x + K 2 x Total Cost = Fuel cost + Insulation cost x = Thickness of insulation F(x) / x = 0 = -K 1 / x 2 + K 2 => x* = {K 1 / K 2 } 1/2 (starred quantities are optimal) Massachusetts Institute of Technology Constrained Optimization Slide 5 of 23 Unconstrained Optimization (5) K 1 = 500 K 2 = 24 X* = 4.56 Optimizing Cost Example Cost Inches of Insulation Fuel Insulation Total Massachusetts Institute of Technology Constrained Optimization Slide 6 of 23 Page 3

4 Constained Optimization 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 than or greater than the constraints (these are upper and lower bounds Massachusetts Institute of Technology Constrained Optimization Slide 7 of 23 Equality Constraints Example: Best use of budget Maximize: Output = Z(X) = a o x 1 a1 x 2 a2 Subject to (s.t.): Total costs = Budget = p 1 x 1 + p 2 x 2 Z(x) Z* Budget Note: Z(X) / X 0 at optimum X Massachusetts Institute of Technology Constrained Optimization Slide 8 of 23 Page 4

5 Approach Constrained Optimization To solve situations of increasing complexity, (for example, those with equality, inequality constraints)... Transform more difficult situation into one we know how to deal with In this case, transform optimization of a constrained situation to optimization of unconstrained situation Massachusetts Institute of Technology Constrained Optimization Slide 9 of 23 Lagrangean Method (1) Transforms equality constraints into unconstrained problem Start with: Opt: F(x) s.t.: g j (x) = b j => g j (x) - b j = 0 Get to: L = F(x) - Σ j λ j [g j (x) - b j ] λj = Lagrangean multipliers (lambdas) -- these are unknown quantities for which we must solve Note: [g j (x) - b j ] = 0 by definition, thus optimum for F(x) = optimum for L Massachusetts Institute of Technology Constrained Optimization Slide 10 of 23 Page 5

6 Lagrangean Method (2) To optimize L: L / x i = 0 i L / λ j = 0 i Example: Opt: F(x) = 6x 1 x 2 s.t.: g(x) = 3x 1 + 4x 2 = 18 L = 6x 1 x 2 - λ(3x 1 + 4x 2-18) L / x 1 = 6x 2-3λ = 0 L / x 2 = 6x 1-4λ = 0 L / λ i = 3x 1 + 4x 2-18 = 0 Massachusetts Institute of Technology Constrained Optimization Slide 11 of 23 Lagrangean Method (3) Solving as unconstrained problem: L / x 1 = 6x 2-3λ = 0 L / x 2 = 6x 1-4λ = 0 L / λ i = 3x 1 + 4x 2-18 = 0 so that: λ = 2x 2 = 1.5x 1 x 2 = 0.75x 1 3x 1 + 3x 1-18 = 0 x 1 * = 18/3 = 6 x 2 * = 18/8 = 2.25 λ* = 4.5 F(x)* = 40.5 Massachusetts Institute of Technology Constrained Optimization Slide 12 of 23 Page 6

7 Shadow Prices Shadow Price is the Rate of change of objective function per unit change of constraint = F(x) / b j This is meaning of Lagrangean multiplier SP j = F(x)*/ b j = λ j Naturally, this is an instantaneous rate The shadow price is extremely important for system design It defines value of changing constraints Massachusetts Institute of Technology Constrained Optimization Slide 13 of 23 Shadow Prices (2) Let s see how this works in example, by changing constraint by 0.1 units: Opt: F(x) = 6x 1 x 2 s.t.: g(x) = 3x 1 + 4x 2 = 18.1 The optimum values of the variables are x 1 * = (18.1)/6 x 2 * = (18.1)/8 Thus F(x)* = 6(18.1/6)(18.1/8) = F(x) = = 0.45 = λ* (0.1) Massachusetts Institute of Technology Constrained Optimization Slide 14 of 23 Page 7

8 Inequality Constraints Example: Housing insulation Min: Costs = K 1 / x + K 2 x s.t.: x 8 (minimum thickness) Optimizing Cost Example Cost Inches of Insulation Fuel Insulation Total Massachusetts Institute of Technology Constrained Optimization Slide 15 of 23 Inequality Constraints (2) Approach: Transform inequalities into equalities, then proceed as before Again, introduce new variable -- the Slack variable that defines slack or distance between constraint and amount used The resulting equations are known as the Kuhn-Tucker conditions Massachusetts Institute of Technology Constrained Optimization Slide 16 of 23 Page 8

9 Inequality Constraints -- insertion of slack variables in Lagrangean A slack variable, s j, for each inequality g j (x) b j => g j (x) + s j 2 = b j g j (x) b j => g j (x) - s j 2 = b j These are squared to be positive start from: opt: F(x) s.t.: g j (x) b j get to: L = F(x) - Σ j λ j [g j (x) + s j 2 - b j ] Massachusetts Institute of Technology Constrained Optimization Slide 17 of 23 Inequality Constraints -- Complementary Slackness Conditions The optimality conditions are: L / x i = 0 L / λ j = 0 plus: L / s j = 2s j λ j = 0 These new equations imply: s j = 0 or λ j 0 s j 0 λ j = 0 They are the complementary slackness conditions. Either slack or lambda =0 i Massachusetts Institute of Technology Constrained Optimization Slide 18 of 23 Page 9

10 Interpretation of Complementary Slackness Conditions Interpretation: If there is slack on b j, (i.e. more than enough of it) => No value to objective function to having more: λ j = F(x) / b j = 0 If λ j 0, then all available b j used => s j = 0 Massachusetts Institute of Technology Constrained Optimization Slide 19 of 23 Application to Example Min: Costs = K 1 / x + K 2 x s.t.: x 8 (minimum thickness) L = K 1 /x + K 2 x - λ[x - s 2 - b] L = 500/x + 24x - λ[x - s 2-8] 500/x λ = 0 2 λs = 0 x - s 2 = 8 If s = 0, x = 8, λ = 31.8 (at that point) Max = Therefore, worth relaxing (in this case, lowering) constraint to get maximum x * = 4.56 Optimum = 221 Massachusetts Institute of Technology Constrained Optimization Slide 20 of 23 Page 10

11 Unconstrained Optimization (5) K 1 = 500 K 2 = 24 X* = 4.56 Optimizing Cost Example Cost Inches of Insulation Fuel Insulation Total Massachusetts Institute of Technology Constrained Optimization Slide 21 of 23 Another application to Example Min: Costs = K 1 / x + K 2 x s.t.: x 4 (NEW MINIMUM) L = K 1 /x + K 2 x - λ[x + s 2 - b] L = 500/x + 24x - λ[x + s 2-4] 500/x λ = 0 2 λs = 0 x - s 2 = 4 If λ = 0, x = 4.56, slack, s 2 = 1.56 Optimum = 221 not worth changing constraint Massachusetts Institute of Technology Constrained Optimization Slide 22 of 23 Page 11

12 Summary of Presentation Important mathematical approaches Lagreangeans Kuhn- Tucker Conditions Important Concept: Shadow Prices THESE ANALYSES GUIDE DESIGNERS TO CHALLENGE CONSTRAINTS Massachusetts Institute of Technology Constrained Optimization Slide 23 of 23 Page 12