Numerical Optimization for Economists

Transcription

1 Numerical Optimization for Economists Todd Munson Mathematics and Computer Science Division Argonne National Laboratory July 18 21, / 104

2 Part I Numerical Optimization I: Static Models 2 / 104

3 Model Formulation where Classify m people into two groups using v variables c {0, 1} m is the known classification d R m v are the observations β R v+1 defines the separator logit distribution function Maximum likelihood problem m max c i log(f (β, d i, )) + (1 c i ) log(1 f (β, d i, )) β i=1 f (β, x) = exp β 0 + v β j x j j=1 1 + exp β 0 + v β j x j j=1 3 / 104

4 Solution Techniques min f (x) x Main ingredients of solution approaches: Local method: given x k (solution guess) compute a step s. Gradient Descent Quasi-Newton Approximation Sequential Quadratic Programming Globalization strategy: converge from any starting point. Trust region Line search 4 / 104

5 Trust-Region Method min s f (x k ) + s T f (x k ) st H(x k )s subject to s k 5 / 104

6 Trust-Region Method 1 Initialize trust-region radius Constant Direction Interpolation 6 / 104

7 Trust-Region Method 1 Initialize trust-region radius Constant Direction Interpolation 2 Compute a new iterate 1 Solve trust-region subproblem min s f (x k ) + s T f (x k ) st H(x k )s subject to s k 7 / 104

8 Trust-Region Method 1 Initialize trust-region radius Constant Direction Interpolation 2 Compute a new iterate 1 Solve trust-region subproblem min s f (x k ) + s T f (x k ) st H(x k )s subject to s k 2 Accept or reject iterate 3 Update trust-region radius Reduction Interpolation 3 Check convergence 8 / 104

9 Solving the Subproblem Moré-Sorensen method Computes global solution to subproblem Conjugate gradient method with trust region Objective function decreases monotonically Some choices need to be made Preconditioner Norm of direction and residual Dealing with negative curvature 9 / 104

10 Line-Search Method min s f (x k ) + s T f (x k ) st (H(x k ) + λ k I )s 10 / 104

11 Line-Search Method 1 Initialize perturbation to zero 2 Solve perturbed quadratic model min s f (x k ) + s T f (x k ) st (H(x k ) + λ k I )s 11 / 104

12 Line-Search Method 1 Initialize perturbation to zero 2 Solve perturbed quadratic model min s f (x k ) + s T f (x k ) st (H(x k ) + λ k I )s 3 Find new iterate 1 Search along Newton direction 2 Search along gradient-based direction 12 / 104

13 Line-Search Method 1 Initialize perturbation to zero 2 Solve perturbed quadratic model min s f (x k ) + s T f (x k ) st (H(x k ) + λ k I )s 3 Find new iterate 1 Search along Newton direction 2 Search along gradient-based direction 4 Update perturbation Decrease perturbation if the following hold Iterative method succeeds Search along Newton direction succeeds Otherwise increase perturbation 5 Check convergence 13 / 104

14 Solving the Subproblem Conjugate gradient method 14 / 104

15 Solving the Subproblem Conjugate gradient method Conjugate gradient method with trust region Initialize radius Constant Direction Interpolation Update radius Reduction Step length Interpolation Some choices need to be made Preconditioner Norm of direction and residual Dealing with negative curvature 15 / 104

16 Performing the Line Search Backtracking Armijo Line search Find t such that f (x k + ts) f (x k ) + σt f (x k ) T s Try t = 1, β, β 2,... for 0 < β < 1 More-Thuente Line search Find t such that f (x k + ts) f (x k ) + σt f (x k ) T s f (x k + ts) T s δ f (x k ) T s Construct cubic interpolant Compute t to minimize interpolant Refine interpolant 16 / 104

17 Updating the Perturbation 1 If increasing and k = 0 k+1 = Proj [l0,u 0 ] 2 If increasing and k > 0 3 If decreasing k+1 = Proj [li,u i ] 4 If k+1 < l d, then k+1 = 0 ( ) α 0 g(x k ) ( max (α i g(x k ), β i k)) k+1 = min (α d g(x k ), β d k) 17 / 104

18 Trust-Region Line-Search Method 1 Initialize trust-region radius Constant Direction Interpolation 2 Compute a new iterate 1 Solve trust-region subproblem min s f (x k ) + s T f (x k ) st H(x k )s subject to s k 2 Search along direction 3 Update trust-region radius Reduction Step length Interpolation 3 Check convergence 18 / 104

19 Iterative Methods Conjugate gradient method Stop if negative curvature encountered Stop if residual norm is small 19 / 104

20 Iterative Methods Conjugate gradient method Stop if negative curvature encountered Stop if residual norm is small Conjugate gradient method with trust region Nash Follow direction to boundary if first iteration Stop at base of direction otherwise Steihaug-Toint Follow direction to boundary Generalized Lanczos Compute tridiagonal approximation Find global solution to approximate problem on boundary Initialize perturbation with approximate minimum eigenvalue 20 / 104

21 Preconditioners No preconditioner Absolute value of Hessian diagonal Absolute value of perturbed Hessian diagonal Incomplete Cholesky factorization of Hessian Block Jacobi with Cholesky factorization of blocks Scaled BFGS approximation to Hessian matrix None Scalar Diagonal of Broyden update Rescaled diagonal of Broyden update Absolute value of Hessian diagonal Absolute value of perturbed Hessian diagonal 21 / 104

22 Norms Residual Preconditioned r M T M 1 Unpreconditioned r 2 Natural r M 1 Direction Preconditioned s M Monotonically increasing s k+1 M > s k M. 22 / 104

23 Norms Residual Preconditioned r M T M 1 Unpreconditioned r 2 Natural r M 1 Direction Preconditioned s M Monotonically increasing s k+1 M > s k M. Unpreconditioned s 2 23 / 104

24 Termination Typical convergence criteria Absolute residual f (x k ) < τ a Relative residual f (x k ) f (x < τ 0) r Unbounded objective f (x k ) < κ Slow progress f (x k ) f (x k 1 ) < ɛ Iteration limit Time limit Solver status 24 / 104

25 Convergence Issues Quadratic convergence best outcome Linear convergence Far from a solution f (x k ) is large Hessian is incorrect disrupts quadratic convergence Hessian is rank deficient f (x k ) is small Limits of finite precision arithmetic 1 f (x k ) converges quadratically to small number 2 f (x k ) hovers around that number with no progress Domain violations such as 1 x when x = 0 Make implicit constraints explicit Nonglobal solution Apply a multistart heuristic Use global optimization solver 25 / 104

26 Some Available Software TRON Newton method with trust-region LBFGS Limited-memory quasi-newton method with line search TAO Toolkit for Advanced Optimization NLS Newton line-search method NTR Newton trust-region method NTL Newton line-search/trust-region method LMVM Limited-memory quasi-newton method CG Nonlinear conjugate gradient methods 26 / 104

27 Model Formulation Economy with n agents and m commodities e R n m are the endowments α R n m and β R n m are the utility parameters λ R n are the social weights Social planning problem ( n m ) α i,k (1 + x i,k ) 1 β i,k max λ i x 0 1 β i,k i=1 k=1 n n subject to x i,k k = 1,..., m i=1 i=1 e i,k 27 / 104

28 Solving Constrained Optimization Problems min f (x) x subject to c(x) 0 Main ingredients of solution approaches: Local method: given x k (solution guess) find a step s. Sequential Quadratic Programming (SQP) Sequential Linear/Quadratic Programming (SLQP) Interior-Point Method (IPM) Globalization strategy: converge from any starting point. Trust region Line search Acceptance criteria: filter or penalty function. 28 / 104

29 Sequential Linear Programming 1 Initialize trust-region radius 2 Compute a new iterate 29 / 104

30 Sequential Linear Programming 1 Initialize trust-region radius 2 Compute a new iterate 1 Solve linear program min s f (x k ) + s T f (x k ) subject to c(x k ) + c(x k ) T s 0 s k 30 / 104

31 Sequential Linear Programming 1 Initialize trust-region radius 2 Compute a new iterate 1 Solve linear program 2 Accept or reject iterate 3 Update trust-region radius 3 Check convergence min s f (x k ) + s T f (x k ) subject to c(x k ) + c(x k ) T s 0 s k 31 / 104

32 Sequential Quadratic Programming 1 Initialize trust-region radius 2 Compute a new iterate 32 / 104

33 Sequential Quadratic Programming 1 Initialize trust-region radius 2 Compute a new iterate 1 Solve quadratic program min s f (x k ) + s T f (x k ) st W (x k )s subject to c(x k ) + c(x k ) T s 0 s k 33 / 104

34 Sequential Quadratic Programming 1 Initialize trust-region radius 2 Compute a new iterate 1 Solve quadratic program min s f (x k ) + s T f (x k ) st W (x k )s subject to c(x k ) + c(x k ) T s 0 s k 2 Accept or reject iterate 3 Update trust-region radius 3 Check convergence 34 / 104

35 Sequential Linear Quadratic Programming 1 Initialize trust-region radius 2 Compute a new iterate 35 / 104

36 Sequential Linear Quadratic Programming 1 Initialize trust-region radius 2 Compute a new iterate 1 Solve linear program to predict active set min d f (x k ) + d T f (x k ) subject to c(x k ) + c(x k ) T d 0 d k 36 / 104

37 Sequential Linear Quadratic Programming 1 Initialize trust-region radius 2 Compute a new iterate 1 Solve linear program to predict active set min d f (x k ) + d T f (x k ) subject to c(x k ) + c(x k ) T d 0 d k 2 Solve equality constrained quadratic program min s f (x k ) + s T f (x k ) st W (x k )s subject to c A (x k ) + c A (x k ) T s = 0 3 Accept or reject iterate 4 Update trust-region radius 3 Check convergence 37 / 104

38 Acceptance Criteria Decrease objective function value: f (x k + s) f (x k ) Decrease constraint violation: c (x k + s) c (x k ) 38 / 104

39 Acceptance Criteria Decrease objective function value: f (x k + s) f (x k ) Decrease constraint violation: c (x k + s) c (x k ) Four possibilities 1 step can decrease both f (x) and c (x) GOOD 2 step can decrease f (x) and increase c (x)??? 3 step can increase f (x) and decrease c (x)??? 4 step can increase both f (x) and c (x) BAD 39 / 104

40 Acceptance Criteria Decrease objective function value: f (x k + s) f (x k ) Decrease constraint violation: c (x k + s) c (x k ) Four possibilities 1 step can decrease both f (x) and c (x) GOOD 2 step can decrease f (x) and increase c (x)??? 3 step can increase f (x) and decrease c (x)??? 4 step can increase both f (x) and c (x) BAD Filter uses concept from multi-objective optimization (h k+1, f k+1 ) dominates (h l, f l ) iff h k+1 h l and f k+1 f l 40 / 104

41 Filter Framework Filter F: list of non-dominated pairs (h l, f l ) new x k+1 is acceptable to filter F iff for all l F 1 h k+1 h l or 2 f k+1 f l 41 / 104

42 Filter Framework Filter F: list of non-dominated pairs (h l, f l ) new x k+1 is acceptable to filter F iff for all l F 1 h k+1 h l or 2 f k+1 f l remove redundant filter entries 42 / 104

43 Filter Framework Filter F: list of non-dominated pairs (h l, f l ) new x k+1 is acceptable to filter F iff for all l F 1 h k+1 h l or 2 f k+1 f l remove redundant filter entries new x k+1 is rejected if for some l F 1 h k+1 > h l and 2 f k+1 > f l 43 / 104

44 Convergence Criteria Feasible and no descent directions Constraint qualification LICQ, MFCQ Linearized active constraints characterize directions Objective gradient is a linear combination of constraint gradients 44 / 104

45 Optimality Conditions If x is a local minimizer and a constraint qualification holds, then there exist multipliers λ 0 such that f (x ) c A (x ) T λ A = 0 Lagrangian function L(x, λ) := f (x) λ T c(x) Optimality conditions can be written as f (x) c(x) T λ = 0 0 λ c(x) 0 Complementarity problem 45 / 104

46 Termination Feasible and complementary min(c(x k ), λ k ) τ f Optimal x L(x k, λ k ) τ o Other possible conditions Slow progress Iteration limit Time limit Multipliers and reduced costs 46 / 104

47 Convergence Issues Quadratic convergence best outcome Globally infeasible linear constraints infeasible Locally infeasible nonlinear constraints locally infeasible Unbounded objective hard to detect Unbounded multipliers constraint qualification not satisfied Linear convergence rate Far from a solution f (x k ) is large Hessian is incorrect disrupts quadratic convergence Hessian is rank deficient f (x k ) is small Limits of finite precision arithmetic Domain violations such as 1 x when x = 0 Make implicit constraints explicit Nonglobal solutions Apply a multistart heuristic Use global optimization solver 47 / 104

48 Some Available Software filtersqp trust-region SQP; robust QP solver filter to promote global convergence SNOPT line-search SQP; null-space CG option l 1 exact penalty function SLIQUE part of KNITRO SLP-EQP trust-region with l 1 penalty use with knitro options = "algorithm=3"; 48 / 104

49 Part II Numerical Optimization II: Optimal Control 49 / 104

50 Model Formulation Maximize discounted utility u( ) is the utility function R is the retirement age T is the terminal age w is the wage β is the discount factor r is the interest rate Optimization problem T β t u(c t ) max s,c t=0 subject to s t+1 = (1 + r)s t + w c t t = 0,..., R 1 s t+1 = (1 + r)s t c t t = R,..., T s 0 = s T +1 = 0 50 / 104

51 Solving Constrained Optimization Problems min f (x) x subject to c(x) 0 Main ingredients of solution approaches: Local method: given x k (solution guess) find a step s. Sequential Quadratic Programming (SQP) Sequential Linear/Quadratic Programming (SLQP) Interior-Point Method (IPM) Globalization strategy: converge from any starting point. Trust region Line search Acceptance criteria: filter or penalty function. 51 / 104

52 Interior-Point Method Reformulate optimization problem with slacks min f (x) x subject to c(x) = 0 x 0 Construct perturbed optimality conditions f (x) c(x) T λ µ F τ (x, y, z) = c(x) X µ τe Central path {x(τ), λ(τ), µ(τ) τ > 0} Apply Newton s method for sequence τ 0 52 / 104

53 Interior-Point Method 1 Compute a new iterate 1 Solve linear system of equations W k c(x k ) T I c(x k ) 0 0 s x s λ = F τ (x k, λ k, µ k ) µ k 0 X k s µ 2 Accept or reject iterate 3 Update parameters 2 Check convergence 53 / 104

54 Convergence Issues Quadratic convergence best outcome Globally infeasible linear constraints infeasible Locally infeasible nonlinear constraints locally infeasible Dual infeasible dual problem is locally infeasible Unbounded objective hard to detect Unbounded multipliers constraint qualification not satisfied Duality gap Domain violations such as 1 x when x = 0 Make implicit constraints explicit Nonglobal solutions Apply a multistart heuristic Use global optimization solver 54 / 104

55 Some Available Software IPOPT open source in COIN-OR line-search filter algorithm KNITRO trust-region Newton to solve barrier problem l 1 penalty barrier function Newton system: direct solves or null-space CG LOQO line-search method Newton system: modified Cholesky factorization 55 / 104

56 Optimal Technology Optimize energy production schedule and transition between old and new reduced-carbon technology to meet carbon targets Maximize social welfare Constraints Limit total greenhouse gas emissions Low-carbon technology less costly as it becomes widespread Assumptions on emission rates, economic growth, and energy costs 56 / 104

57 Model Formulation Finite time: t [0, T ] Instantaneous energy output: q o (t) and q n (t) Cumulative energy output: x o (t) and x n (t) x n (t) = t 0 q n (τ)dτ Discounted greenhouse gases emissions T 0 e at (b o q o (t) + b n q n (t)) dt z T Consumer surplus S(Q(t), t) derived from utility Production costs c o per unit cost of old technology c n (x n (t)) per unit cost of new technology (learning by doing) 57 / 104

58 Continuous-Time Model max {q o,q n,x n,z}(t) subject to T 0 e rt [S(q o (t) + q n (t), t) c o q o (t) c n (x n (t))q n (t)] dt x n (t) = q n (t) x(0) = x 0 = 0 ż(t) = e at (b o q o (t) + b n q n (t)) z(0) = z 0 = 0 z(t ) z T q o (t) 0, q n (t) / 104

59 Optimal Technology Penetration Discretization: t [0, T ] replaced by N + 1 equally spaced points t i = ih h := T /N time integration step-length approximate qi n q n (t i ) etc. Replace differential equation ẋ(t) = q n (t) by x i+1 = x i + hqi n 59 / 104

60 Optimal Technology Penetration Discretization: t [0, T ] replaced by N + 1 equally spaced points t i = ih h := T /N time integration step-length approximate qi n q n (t i ) etc. Replace differential equation ẋ(t) = q n (t) by x i+1 = x i + hqi n Output of new technology between t = 24 and t = / 104

61 Solution with Varying h Output for different discretization schemes and step-sizes 61 / 104

62 Optimal Technology Penetration Add adjustment cost to model building of capacity: Capital and Investment: Notation: K j (t) amount of capital in technology j at t. I j (t) investment to increase K j (t). initial capital level as K j 0 : Q(t) = q o (t) + q n (t) C(t) = C o (q o (t), K o (t)) + C n (q n (t), K n (t)) I (t) = I o (t) + I n (t) K(t) = K o (t) + K n (t) 62 / 104

63 Optimal Technology Penetration maximize {q j,k j,i j,x,z}(t) { T 0 [ S(Q(t), ] } e rt t) C(t) K(t) dt + e rt K(T ) subject to ẋ(t) = q n (t), x(0) = x 0 = 0 K j (t) = δk j (t) + I j (t), K j (0) = K j 0, j {o, n} ż(t) = e at [b o q o (t) + b n q n (t)], z(0) = z 0 = 0 z(t ) z T q j (t) 0, j {o, n} I j (t) 0, j {o, n} 63 / 104

64 Optimal Technology Penetration Optimal output, investment, and capital for 50% CO2 reduction. 64 / 104

65 Pitfalls of Discretizations [Hager, 2000] Optimal Control Problem minimize 1 2 subject to 1 0 u 2 (t) + 2y 2 (t)dt ẏ(t) = 1 2y(t) + u(t), t [0, 1], y(0) = 1. y (t) = 2e3t + e 3 e 3t/2 (2 + e 3 ), u (t) = 2(e3t e 3 ) e 3t/2 (2 + e 3 ). 65 / 104

66 Pitfalls of Discretizations [Hager, 2000] Optimal Control Problem minimize 1 2 subject to 1 0 u 2 (t) + 2y 2 (t)dt ẏ(t) = 1 2y(t) + u(t), t [0, 1], y(0) = 1. Discretize with 2nd order RK minimize h 2 K 1 k=0 u 2 k+1/2 +2y 2 k+1/2 subject to (k = 0,..., K): y k+1/2 = y k + h 2 ( 1 2 y k + u k ), y k+1 = y k + h( 1 2 y k+1/2 + u k+1/2 ), y (t) = 2e3t + e 3 e 3t/2 (2 + e 3 ), u (t) = 2(e3t e 3 ) e 3t/2 (2 + e 3 ). 66 / 104

67 Pitfalls of Discretizations [Hager, 2000] Optimal Control Problem minimize 1 2 subject to 1 0 u 2 (t) + 2y 2 (t)dt ẏ(t) = 1 2y(t) + u(t), t [0, 1], y(0) = 1. y (t) = 2e3t + e 3 e 3t/2 (2 + e 3 ), u (t) = 2(e3t e 3 ) e 3t/2 (2 + e 3 ). Discretize with 2nd order RK minimize h 2 K 1 k=0 u 2 k+1/2 +2y 2 k+1/2 subject to (k = 0,..., K): y k+1/2 = y k + h 2 ( 1 2 y k + u k ), y k+1 = y k + h( 1 2 y k+1/2 + u k+1/2 ), Discrete solution (k = 0,..., K): y k = 1, y k+1/2 = 0, u k = 4 + h 2h, u k+1/2 = 0, DOES NOT CONVERGE! 67 / 104

68 Tips to Solve Continuous-Time Problems Use discretize-then-optimize with different schemes Refine discretization: h = 1 discretization is nonsense Check implied discretization of adjoints 68 / 104

69 Tips to Solve Continuous-Time Problems Use discretize-then-optimize with different schemes Refine discretization: h = 1 discretization is nonsense Check implied discretization of adjoints Alternative: Optimize-Then-Discretize Consistent adjoint/dual discretization Discretized gradients can be wrong! Harder for inequality constraints 69 / 104

70 Part III Numerical Optimization III: Complementarity Constraints 70 / 104

71 Nash Games Non-cooperative game played by n individuals Each player selects a strategy to optimize their objective Strategies for the other players are fixed Equilibrium reached when no improvement is possible 71 / 104

72 Nash Games Non-cooperative game played by n individuals Each player selects a strategy to optimize their objective Strategies for the other players are fixed Equilibrium reached when no improvement is possible Characterization of two player equilibrium (x, y ) { arg min f x 1 (x, y ) x 0 { subject to c 1 (x) 0 arg min f 2 (x, y) y y 0 subject to c 2 (y) 0 72 / 104

73 Nash Games Non-cooperative game played by n individuals Each player selects a strategy to optimize their objective Strategies for the other players are fixed Equilibrium reached when no improvement is possible Characterization of two player equilibrium (x, y ) { arg min f x 1 (x, y ) x 0 { subject to c 1 (x) 0 arg min f 2 (x, y) y y 0 subject to c 2 (y) 0 Many applications in economics Bimatrix games Cournot duopoly models General equilibrium models Arrow-Debreau models 73 / 104

74 Complementarity Formulation Assume each optimization problem is convex f 1 (, y) is convex for each y f 2 (x, ) is convex for each x c 1 ( ) and c 2 ( ) satisfy constraint qualification Then the first-order conditions are necessary and sufficient min f 1(x, y ) x 0 subject to c 1(x) 0 0 x xf1(x, y ) + λ T 1 xc 1(x) 0 0 λ 1 c 1(x) 0 74 / 104

75 Complementarity Formulation Assume each optimization problem is convex f 1 (, y) is convex for each y f 2 (x, ) is convex for each x c 1 ( ) and c 2 ( ) satisfy constraint qualification Then the first-order conditions are necessary and sufficient min f 2(x, y) y 0 subject to c 2(y) 0 0 y y f2(x, y) + λ T 2 y c 2(y) 0 0 λ 2 c 2(y) 0 75 / 104

76 Complementarity Formulation Assume each optimization problem is convex f 1 (, y) is convex for each y f 2 (x, ) is convex for each x c 1 ( ) and c 2 ( ) satisfy constraint qualification Then the first-order conditions are necessary and sufficient 0 x x f 1 (x, y) + λ T 1 xc 1 (x) 0 0 y y f 2 (x, y) + λ T 2 y c 2 (y) 0 0 λ 1 c 1 (y) 0 0 λ 2 c 2 (y) 0 Nonlinear complementarity problem Square system number of variables and constraints the same Each solution is an equilibrium for the Nash game 76 / 104

77 Model Formulation Economy with n agents and m commodities e R n m are the endowments α R n m and β R n m are the utility parameters p R m are the commodity prices Agent i maximizes utility with budget constraint max x i, 0 subject to m α i,k (1 + x i,k ) 1 β i,k 1 β i,k m p k (x i,k e i,k ) 0 k=1 k=1 Market k sets price for the commodity 0 p k n (e i,k x i,k ) 0 i=1 77 / 104

78 Newton Method for Nonlinear Equations 78 / 104

79 Newton Method for Nonlinear Equations 79 / 104

80 Newton Method for Nonlinear Equations 80 / 104

81 Newton Method for Nonlinear Equations 81 / 104

82 Methods for Complementarity Problems Sequential linearization methods (PATH) 1 Solve the linear complementarity problem 0 x F (x k ) + F (x k )(x x k ) 0 2 Perform a line search along merit function 3 Repeat until convergence 82 / 104

83 Methods for Complementarity Problems Sequential linearization methods (PATH) 1 Solve the linear complementarity problem 0 x F (x k ) + F (x k )(x x k ) 0 2 Perform a line search along merit function 3 Repeat until convergence Semismooth reformulation methods (SEMI) Solve linear system of equations to obtain direction Globalize with a trust region or line search Less robust in general Interior-point methods 83 / 104

84 Semismooth Reformulation Define Fischer-Burmeister function φ(a, b) := a + b a 2 + b 2 φ(a, b) = 0 iff a 0, b 0, and ab = 0 Define the system [Φ(x)] i = φ(x i, F i (x)) x solves complementarity problem iff Φ(x ) = 0 Nonsmooth system of equations 84 / 104

85 Semismooth Algorithm 1 Calculate H k B Φ(x k ) and solve the following system for d k : H k d k = Φ(x k ) If this system either has no solution, or Ψ(x k ) T d k p 1 d k p 2 is not satisfied, let d k = Ψ(x k ). 85 / 104

86 Semismooth Algorithm 1 Calculate H k B Φ(x k ) and solve the following system for d k : H k d k = Φ(x k ) If this system either has no solution, or Ψ(x k ) T d k p 1 d k p 2 is not satisfied, let d k = Ψ(x k ). 2 Compute smallest nonnegative integer i k such that Ψ(x k + β i k d k ) Ψ(x k ) + σβ i k Ψ(x k )d k 3 Set x k+1 = x k + β i k d k, k = k + 1, and go to / 104

87 Convergence Issues Quadratic convergence best outcome Linear convergence Far from a solution r(x k ) is large Jacobian is incorrect disrupts quadratic convergence Jacobian is rank deficient r(x k ) is small Converge to local minimizer guarantees rank deficiency Limits of finite precision arithmetic 1 r(x k ) converges quadratically to small number 2 r(x k ) hovers around that number with no progress Domain violations such as 1 x when x = 0 87 / 104

88 Some Available Software PATH sequential linearization method MILES sequential linearization method SEMI semismooth linesearch method TAO Toolkit for Advanced Optimization SSLS full-space semismooth linesearch methods ASLS active-set semismooth linesearch methods RSCS reduced-space method 88 / 104

89 Definition Leader-follower game Dominant player (leader) selects a strategy y Then followers respond by playing a Nash game x i { arg min x i 0 f i (x, y) subject to c i (x i ) 0 Leader solves optimization problem with equilibrium constraints min y 0,x,λ g(x, y) subject to h(y) 0 0 x i xi f i (x, y) + λ T i xi c i (x i ) 0 0 λ i c i (x i ) 0 Many applications in economics Optimal taxation Tolling problems 89 / 104

90 Model Formulation Economy with n agents and m commodities e R n m are the endowments α R n m and β R n m are the utility parameters p R m are the commodity prices Agent i maximizes utility with budget constraint max x i, 0 subject to m α i,k (1 + x i,k ) 1 β i,k 1 β i,k m p k (x i,k e i,k ) 0 k=1 k=1 Market k sets price for the commodity 0 p k n (e i,k x i,k ) 0 i=1 90 / 104

91 Nonlinear Programming Formulation min x,y,λ,s,t 0 g(x, y) subject to h(y) 0 s i = xi f i (x, y) + λ T i xi c i (x i ) t i = c i (x i ) ) (si T x i + λ i t i 0 i Constraint qualification fails Lagrange multiplier set unbounded Constraint gradients linearly dependent Central path does not exist Able to prove convergence results for some methods Reformulation very successful and versatile in practice 91 / 104

92 Penalization Approach min g(x, y) + π ) (si T x i + λ i t i x,y,λ,s,t 0 i subject to h(y) 0 s i = xi f i (x, y) + λ T i xi c i (x i ) t i = c i (x i ) Optimization problem satisfies constraint qualification Need to increase π 92 / 104

93 Relaxation Approach min x,y,λ,s,t 0 g(x, y) subject to h(y) 0 s i = xi f i (x, y) + λ T i xi c i (x i ) t i = c i (x i ) Need to decrease τ ) (si T x i + λ i t i τ i 93 / 104

94 Limitations Multipliers may not exist Solvers can have a hard time computing solutions Try different algorithms Compute feasible starting point Stationary points may have descent directions Checking for descent is an exponential problem Strong stationary points found in certain cases Many stationary points global optimization 94 / 104

95 Limitations Multipliers may not exist Solvers can have a hard time computing solutions Try different algorithms Compute feasible starting point Stationary points may have descent directions Checking for descent is an exponential problem Strong stationary points found in certain cases Many stationary points global optimization Formulation of follower problem Multiple solutions to Nash game Nonconvex objective or constraints Existence of multipliers 95 / 104

96 Part IV Numerical Optimization IV: Extensions 96 / 104

97 Global Optimization I need to find the GLOBAL minimum! use any NLP solver (often work well!) use the multi-start trick from previous slides global optimization based on branch-and-reduce: BARON constructs global underestimators refines region by branching tightens bounds by solving LPs solve problems with 100s of variables voodoo solvers: genetic algorithm & simulated annealing no convergence theory... usually worse than deterministic 97 / 104

98 Derivative-Free Optimization My model does not have derivatives! Change your model... good models have derivatives! pattern-search methods for min f (x) evaluate f (x) at stencil x k + M move to new best point extend to NLP; some convergence theory h matlab: NOMADm.m; parallel APPSPACK solvers based on building interpolating quadratic models DFO project on Mike Powell s NEWUOA quadratic model voodoo solvers: genetic algorithm & simulated annealing no convergence theory... usually worse than deterministic 98 / 104

99 Optimization with Integer Variables Mixed-Integer Nonlinear Program (MINLP) modeling discrete choices 0 1 variables modeling integer decisions integer variables e.g. number of different stocks in portfolio (8-10) not number of beers sold at Goose Island (millions) MINLP solvers: branch (separate z i = 0 and z i = 1) and cut solve millions of NLP relaxations: MINLPBB, SBB outer approximation: iterate MILP and NLP solvers BONMIN (COIN-OR) & FilMINT on NEOS 99 / 104

100 Portfolio Management N: Universe of asset to purchase x i : Amount of asset i to hold B: Budget minimize u(x) subject to i N x i = B, x / 104

101 Portfolio Management N: Universe of asset to purchase x i : Amount of asset i to hold B: Budget minimize u(x) subject to i N x i = B, x 0 Markowitz: u(x) def = α T x + λx T Qx α: maximize expected returns Q: variance-covariance matrix of expected returns λ: minimize risk; aversion parameter 101 / 104

102 More Realistic Models b R N of benchmark holdings Benchmark Tracking: u(x) def = (x b) T Q(x b) Constraint on E[Return]: α T x r 102 / 104

103 More Realistic Models b R N of benchmark holdings Benchmark Tracking: u(x) def = (x b) T Q(x b) Constraint on E[Return]: α T x r Limit Names: i N : x i > 0 K Use binary indicator variables to model the implication x i > 0 y i = 1 Implication modeled with variable upper bounds: i N y i K x i By i i N 103 / 104

104 Optimization Conclusions Optimization is General Modeling Paradigm linear, nonlinear, equations, inequalities integer variables, equilibrium, control AMPL (GAMS) Modeling and Programming Languages express optimization problems use automatic differentiation easy access to state-of-the-art solvers Optimization Software open-source: COIN-OR, IPOPT, SOPLEX, & ASTROS (soon) current solver limitations on laptop: 1,000,000 variables/constraints for LPs 100,000 variables/constraints for NLPs/NCPs 100 variables/constraints for global optimization 500,000,000 variable LP on BlueGene/P 104 / 104