Aone-phaseinteriorpointmethodfornonconvexoptimization

Transcription

1 Aone-phaseinteriorpointmethodfornonconvexoptimization Oliver Hinder, Yinyu Ye Department of Management Science and Engineering Stanford University April 28, 2018

2 Sections 1 Motivation 2 Why a one-phase method is desirable 3 Our one-phase IPM 4 Theory 5 Numerical results

3 Motivation

4 Motivation 1 Finding KKT points is an important problem 2 Detecting infeasibility is important

5 The problem Find KKT point using an interior point method: min f (x) x2r d subject to a(x) apple 0 where f : R d! R and a : R d! R m are di erentiable.

6 The problem Find KKT point using an interior point method: min f (x) x2r d subject to a(x) apple 0 where f : R d! R and a : R d! R m are di erentiable f (x) x x

7 Local solvers I Second-order methods. I Active set methods [MINOS, SNOPT]. Use on: sparse problems with few degrees of freedom, warm starting capabilities. Robust. I Interior point methods (IPMs) [IPOPT, KNITRO, LOQO]. Use on: huge-sparse problems. I First-order methods [LANCELOT]. Use on: well-conditioned or poor sparsity structure.

8 Real applications using local solvers I Engineering design I Orion heat shield design I High Speed Civil Transport (not commissioned) I 1995 America s cup (sail boat design) I Trajectory optimization I International Space Station maneuvered using zero fuel I Network operation I I I Drinking water Electricity Gas KKT points seem to be good solutions.

9 Real applications using local solvers I Engineering design I Orion heat shield design I High Speed Civil Transport (not commissioned) I 1995 America s cup (sail boat design) I Trajectory optimization I International Space Station maneuvered using zero fuel I Network operation I I I Drinking water Electricity Gas KKT points seem to be good solutions.

10 Real applications using local solvers I Engineering design I Orion heat shield design I High Speed Civil Transport (not commissioned) I 1995 America s cup (sail boat design) I Trajectory optimization I International Space Station maneuvered using zero fuel I Network operation I I I Drinking water Electricity Gas KKT points seem to be good solutions.

11 Drinking water network operation I Objective: minimizing operating cost. I Decision variables: pump levels I Constraints: meet demands and satisfy physics

12 Drinking water network operation I Objective: minimizing operating cost. I Decision variables: pump levels I Constraints: meet demands and satisfy physics Comments: I Constraints are nonconvex! I Problem is huge-scale (think of the number of pipes in a city) I Nice sparsity structure = use second-order method I User experiments with closing pipes! lots of problem instances that might be infeasible! I Similar issues for optimal power flow

13 Drinking water network operation I Objective: minimizing operating cost. I Decision variables: pump levels I Constraints: meet demands and satisfy physics Comments: I Constraints are nonconvex! I Problem is huge-scale (think of the number of pipes in a city) I Nice sparsity structure = use second-order method I User experiments with closing pipes! lots of problem instances that might be infeasible! I Similar issues for optimal power flow Takeaway: nonconvex optimization and infeasibility detection are important.

14 Why a one-phase method?

15 Why a one-phase method? 1 Best IPM(s) for linear programming are one-phase methods 2 Challenging to develop a one-phase nonconvex IPM

16 Problem we wish to solve: optimal infeasible

17 Problem we wish to solve: optimal infeasible min f (x) x2r d subject to a(x) apple 0

18 Problem we wish to solve: optimal min f (x) x2r d subject to a(x) apple 0 infeasible min max x2r d i a i (x)

19 Abridged history of IPM for linear programming

20 Abridged history of IPM for linear programming 1 Primal-dual feasible start [Monteiro and Adler, 1989]

21 Abridged history of IPM for linear programming 1 Primal-dual feasible start [Monteiro and Adler, 1989] 2 Phase-one, phase-two method

22 Abridged history of IPM for linear programming 1 Primal-dual feasible start [Monteiro and Adler, 1989] 2 Phase-one, phase-two method 3 Penalty method [McShane, Monma, and Shanno, 1989]

23 Abridged history of IPM for linear programming 1 Primal-dual feasible start [Monteiro and Adler, 1989] 2 Phase-one, phase-two method 3 Penalty method [McShane, Monma, and Shanno, 1989] 4 One-phase methods I Infeasible start IPM [Lustig [1990], Mehrotra [1992]] I Homogenous algorithm [Ye, Todd, and Mizuno [1994]]

24 Abridged history of IPM for linear programming 1 Primal-dual feasible start [Monteiro and Adler, 1989] 2 Phase-one, phase-two method 3 Penalty method [McShane, Monma, and Shanno, 1989] 4 One-phase methods I Infeasible start IPM [Lustig [1990], Mehrotra [1992]] I Homogenous algorithm [Ye, Todd, and Mizuno [1994]] Extended to convex [Andersen and Ye [1999]] and conic [Sturm [2002], Andersen, Roos, and Terlaky [2003]].

25 Abridged history of IPM for linear programming 1 Primal-dual feasible start [Monteiro and Adler, 1989] 2 Phase-one, phase-two method 3 Penalty method [McShane, Monma, and Shanno, 1989] 4 One-phase methods I Infeasible start IPM [Lustig [1990], Mehrotra [1992]] I Homogenous algorithm [Ye, Todd, and Mizuno [1994]] Extended to convex [Andersen and Ye [1999]] and conic [Sturm [2002], Andersen, Roos, and Terlaky [2003]]. Generalize IPM to nonconvex optimization: 1 Homogenous algorithm? 2 Lustig s IPM?

26 Abridged history of IPM for linear programming 1 Primal-dual feasible start [Monteiro and Adler, 1989] 2 Phase-one, phase-two method 3 Penalty method [McShane, Monma, and Shanno, 1989] 4 One-phase methods I Infeasible start IPM [Lustig [1990], Mehrotra [1992]] I Homogenous algorithm [Ye, Todd, and Mizuno [1994]] Extended to convex [Andersen and Ye [1999]] and conic [Sturm [2002], Andersen, Roos, and Terlaky [2003]]. Generalize IPM to nonconvex optimization: 1 Homogenous algorithm? 2 Lustig s IPM?

27 Failure of infeasible start IPM for nonconvex optimization Wächter and Biegler [2000] show an infeasible start IPM [Lustig, 1990] on: min x x 2 1 x 1, converges to infeasible point that is not a (local) certificate of infeasibility.

28 Solutions to this problem

29 Solutions to this problem I Two phases [IPOPT, Wächter and Biegler [2006]] I Main phase and feasibility restoration phase I Poor performance on infeasible problems

30 Solutions to this problem I Two phases [IPOPT, Wächter and Biegler [2006]] I Main phase and feasibility restoration phase I Poor performance on infeasible problems I Penalty (or big-m) method [Chen and Goldfarb [2006], Curtis [2012]] I Big penalties = slow convergence! Small penalties = failure!

31 Solutions to this problem I Two phases [IPOPT, Wächter and Biegler [2006]] I Main phase and feasibility restoration phase I Poor performance on infeasible problems I Penalty (or big-m) method [Chen and Goldfarb [2006], Curtis [2012]] I Big penalties = slow convergence! Small penalties = failure! I Compute two directions [KNITRO, Byrd, Nocedal, and Waltz [2006]] I Directions: 1 minimize constraint violation 2 improve optimality I Solve two di erent linear systems (with two di erent matrices) at each step. None of these methods reduce to a successful LP IPM!

32 Solutions to this problem I Two phases [IPOPT, Wächter and Biegler [2006]] I Main phase and feasibility restoration phase I Poor performance on infeasible problems I Penalty (or big-m) method [Chen and Goldfarb [2006], Curtis [2012]] I Big penalties = slow convergence! Small penalties = failure! I Compute two directions [KNITRO, Byrd, Nocedal, and Waltz [2006]] I Directions: 1 minimize constraint violation 2 improve optimality I Solve two di erent linear systems (with two di erent matrices) at each step. None of these methods reduce to a successful LP IPM! Why does Lustig s IPM fail?

33 What goes wrong with typical IPMs? IPMs generate sequence of approximate local minimizers to shifted log barrier: min f (x) µ X i log(r i a i (x)). Primal residual r implicitly defined (r = a(x)+s). Consider following example:

34 What goes wrong with typical IPMs? IPMs generate sequence of approximate local minimizers to shifted log barrier: min f (x) µ X i log(r i a i (x)). Primal residual r implicitly defined (r = a(x)+s). Inwards and outwards shift = bad!! " > $! % < $

35 What goes wrong with typical IPMs? IPMs generate sequence of approximate local minimizers to shifted log barrier: min f (x) µ X i log(r i a i (x)). Primal residual r implicitly defined (r = a(x)+s). Newton step fails!

36 What goes wrong with typical IPMs? IPMs generate sequence of approximate local minimizers to shifted log barrier: min f (x) µ X i log(r i a i (x)). Primal residual r implicitly defined (r = a(x)+s). We always shift constraints outwards:! % > $! " > $

37 Our one-phase IPM

38 Pictorial representation of our algorithm Recall: min f (x) µ X i log(r i a i (x)), with r i = µ 0. Shift and barrier parameter same size!!

39 Pictorial representation of our algorithm At each iteration: If shifted log barrier KKT then Aggressive step: reduce µ Else Stabilization step: min shifted log barrier End

40 Pictorial representation of our algorithm At each iteration: If shifted log barrier KKT then Aggressive step: reduce µ Else Stabilization step: min shifted log barrier End

41 Overview of key algorithm properties 1 Di erences between nonconvex programming literature are (1) Reduce constraint violation + µ at same rate (2) Nonlinear update slack variables (3) Carefully initialize slack variables 2 For LP reduces to IPM of style of Lustig [1990], Mehrotra [1992].

42 Primal-dual direction computation KKT system of log barrier + Newton s method ) IPM direction.

43 Primal-dual direction computation KKT system of log barrier + Newton s method ) IPM direction. For example, LOQO [Vanderbei, 1999]: r 2 xxl(x k, y k )+ k I ra(x k ) T 0 d 4 ra(x k x k ) 0 I 5 4d k 5 0 S k Y k y = 4 ds k 3 (rf (x k )+ra(x k ) T y k ) (a(x k )+s k ) 5 k µ k e Y k s k where k 2 [0, 1] is reduction factor in µ k.

44 Primal-dual direction computation KKT system of log barrier + Newton s method ) IPM direction. Our primal-dual direction: r 2 xxl(x k, y k )+ k I ra(x k ) T 0 d 4 ra(x k x k ) 0 I 5 4d k 5 0 S k Y k y = 4 ds k 3 (rf (x k )+ra(x k ) T y k ) (1 k )(a(x k )+s k ) 5 k µ k e Y k s k where k 2 [0, 1] is reduction factor in µ k and constraint violation (1). Common strategy in LP!

45 Primal-dual direction computation KKT system of log barrier + Newton s method ) IPM direction. Our primal-dual direction: r 2 xxl(x k, y k )+ k I ra(x k ) T 0 d 4 ra(x k x k ) 0 I 5 4d k 5 0 S k Y k y = 4 ds k 3 (rf (x k )+ra(x k ) T y k ) (1 k )(a(x k )+s k ) 5 k µ k e Y k s k where k 2 [0, 1] is reduction factor in µ k and constraint violation (1). Common strategy in LP! We set k = 1 for stabilization steps, k < 1 for aggressive steps.

46 Iterate update For step size k 2 [0, 1] update iterates as follows: µ k+1 µ k + k d k µ x k+1 y k+1 s k+1 x k + k d k x y k + k d k y s k + k d k s

47 Iterate update For step size k 2 [0, 1] update iterates as follows: µ k+1 µ k + k d k µ x k+1 y k+1 x k + k d k x y k + k dy k h (((((((( s k+1 hhhhhhh s k + k ds k! s k+1 k dµ k 1 µ k (a(x k )+s k ) a(x k+1 ) (2)

48 Iterate update For step size k 2 [0, 1] update iterates as follows: µ k+1 µ k + k d k µ x k+1 y k+1 x k + k d k x y k + k dy k h (((((((( s k+1 hhhhhhh s k + k ds k! s k+1 k dµ k 1 µ k (a(x k )+s k ) a(x k+1 ) (2) If we start with: then: a(x 1 )+s 1 = µ 1 e a(x k )+s k = µ k e (3) implicitly satisfied in Mehrotra [1992] IPM for linear programming.

49 Iterate update For step size k 2 [0, 1] update iterates as follows: µ k+1 µ k + k d k µ x k+1 y k+1 x k + k d k x y k + k dy k h (((((((( s k+1 hhhhhhh s k + k ds k! s k+1 k dµ k 1 µ k (a(x k )+s k ) a(x k+1 ) (2) If we start with: then: a(x 1 )+s 1 = µ 1 e a(x k )+s k = µ k e (3) implicitly satisfied in Mehrotra [1992] IPM for linear programming.

50 Iterate update For step size k 2 [0, 1] update iterates as follows: µ k+1 µ k + k d k µ x k+1 y k+1 x k + k d k x y k + k dy k h (((((((( s k+1 hhhhhhh s k + k ds k! s k+1 k dµ k 1 µ k (a(x k )+s k ) a(x k+1 )= s k + k d k s {z } when a is linear (2) If we start with: then: a(x 1 )+s 1 = µ 1 e a(x k )+s k = µ k e (3) implicitly satisfied in Mehrotra [1992] IPM for linear programming.

51 Recap: our one phase IPM 1 Di erences between nonconvex programming literature are (1) Reduce constraint violation + µ at same rate (2) Nonlinear update slack variables (3) Carefully initialize slack variables 2 For LP reduces to IPM of style of Lustig [1990], Mehrotra [1992].

52 Theory

53 Theory 1 The algorithm terminates at KKT point (for optimality, infeasibility, or unboundedness). 2 The dual multipliers are bounded under general conditions.

54 The algorithm (eventually) terminates Assume that the objective and constraints are di erentiable. Theorem ([Hinder and Ye, 2018]) After a finite number of iterations the one-phase algorithm terminates with either:

55 The algorithm (eventually) terminates Assume that the objective and constraints are di erentiable. Theorem ([Hinder and Ye, 2018]) After a finite number of iterations the one-phase algorithm terminates with either: I an approximate (scaled) KKT solution

56 The algorithm (eventually) terminates Assume that the objective and constraints are di erentiable. Theorem ([Hinder and Ye, 2018]) After a finite number of iterations the one-phase algorithm terminates with either: I an approximate (scaled) KKT solution I an approximate KKT solution to the problem of minimizing the (infinity norm) feasibility violation

57 The algorithm (eventually) terminates Assume that the objective and constraints are di erentiable. Theorem ([Hinder and Ye, 2018]) After a finite number of iterations the one-phase algorithm terminates with either: I an approximate (scaled) KKT solution I an approximate KKT solution to the problem of minimizing the (infinity norm) feasibility violation I an approximate certificate of the (shifted) feasible region being unbounded

58 Proof idea Approximately solve shifted log barrier sub-problem with point (x, s, y, µ).

59 Proof idea Approximately solve shifted log barrier sub-problem with point (x, s, y, µ). Let C 1/. At this point there are two possibilities:

60 Proof idea Approximately solve shifted log barrier sub-problem with point (x, s, y, µ). Let C 1/. At this point there are two possibilities: 1 s µ/c

61 Proof idea Approximately solve shifted log barrier sub-problem with point (x, s, y, µ). Let C 1/. At this point there are two possibilities: 1 s µ/c 2 9i s.t. s i apple µ/c

62 Proof idea Approximately solve shifted log barrier sub-problem with point (x, s, y, µ). Let C 1/. At this point there are two possibilities: 1 s µ/c I We are not too close to boundary ) can reduce the constraint violation. 2 9i s.t. s i apple µ/c

63 Proof idea Approximately solve shifted log barrier sub-problem with point (x, s, y, µ). Let C 1/. At this point there are two possibilities: 1 s µ/c I We are not too close to boundary ) can reduce the constraint violation. 2 9i s.t. s i apple µ/c

64 Proof idea Approximately solve shifted log barrier sub-problem with point (x, s, y, µ). Let C 1/. At this point there are two possibilities: 1 s µ/c I We are not too close to boundary ) can reduce the constraint violation. 2 9i s.t. s i apple µ/c I Since si y i µ we get y i ' C.

65 Proof idea Approximately solve shifted log barrier sub-problem with point (x, s, y, µ). Let C 1/. At this point there are two possibilities: 1 s µ/c I We are not too close to boundary ) can reduce the constraint violation. 2 9i s.t. s i apple µ/c I Since si y i µ we get y i ' C. I The (scaled) dual variables give an approximate KKT solution to minimizing the constraint violation.

66 Proof idea Approximately solve shifted log barrier sub-problem with point (x, s, y, µ). Let C 1/. At this point there are two possibilities: 1 s µ/c I We are not too close to boundary ) can reduce the constraint violation. 2 9i s.t. s i apple µ/c I Since si y i µ we get y i ' C. I The (scaled) dual variables give an approximate KKT solution to minimizing the constraint violation. I Critically we maintain a(x k )+s k 0.

67 Proof idea Approximately solve shifted log barrier sub-problem with point (x, s, y, µ). Let C 1/. At this point there are two possibilities: 1 s µ/c I We are not too close to boundary ) can reduce the constraint violation. 2 9i s.t. s i apple µ/c I Since si y i µ we get y i ' C. I The (scaled) dual variables give an approximate KKT solution to minimizing the constraint violation. I Critically we maintain a(x k )+s k 0. I Example of Wächter and Biegler [2000] IPMs go to a point with a 1(x)+s 1 < 0 a 2(x)+s 2 > 0.

68 Boundedness of dual multipliers The algorithm generates a sequence dual multipliers y k. If ky k k!1then numerical issues!

69 Boundedness of dual multipliers The algorithm generates a sequence dual multipliers y k. If ky k k!1then numerical issues! Theorem ([Haeser, Hinder, and Ye, 2017]) Assume that

70 Boundedness of dual multipliers The algorithm generates a sequence dual multipliers y k. If ky k k!1then numerical issues! Theorem ([Haeser, Hinder, and Ye, 2017]) Assume that I primal iterates are converging to (x, s )

71 Boundedness of dual multipliers The algorithm generates a sequence dual multipliers y k. If ky k k!1then numerical issues! Theorem ([Haeser, Hinder, and Ye, 2017]) Assume that I primal iterates are converging to (x, s ) I (x, s ) is a KKT point

72 Boundedness of dual multipliers The algorithm generates a sequence dual multipliers y k. If ky k k!1then numerical issues! Theorem ([Haeser, Hinder, and Ye, 2017]) Assume that I primal iterates are converging to (x, s ) I (x, s ) is a KKT point I su cient conditions for local optimality hold at (x, s ) (e.g. convexity)

73 Boundedness of dual multipliers The algorithm generates a sequence dual multipliers y k. If ky k k!1then numerical issues! Theorem ([Haeser, Hinder, and Ye, 2017]) Assume that I primal iterates are converging to (x, s ) I (x, s ) is a KKT point I su cient conditions for local optimality hold at (x, s ) (e.g. convexity) Then a subsequence of the dual multipliers is bounded.

74 Boundedness of dual multipliers The algorithm generates a sequence dual multipliers y k. If ky k k!1then numerical issues! Theorem ([Haeser, Hinder, and Ye, 2017]) Assume that I primal iterates are converging to (x, s ) I (x, s ) is a KKT point I su cient conditions for local optimality hold at (x, s ) (e.g. convexity) Then a subsequence of the dual multipliers is bounded. I Similar result already known for linear programming [Mizuno, Todd, and Ye, 1995].

75 Boundedness of dual multipliers The algorithm generates a sequence dual multipliers y k. If ky k k!1then numerical issues! Theorem ([Haeser, Hinder, and Ye, 2017]) Assume that I primal iterates are converging to (x, s ) I (x, s ) is a KKT point I su cient conditions for local optimality hold at (x, s ) (e.g. convexity) Then a subsequence of the dual multipliers is bounded. I Similar result already known for linear programming [Mizuno, Todd, and Ye, 1995]. I IPOPT ky k k!1[haeser, Hinder, and Ye, 2017].

76 Numerical results

77 Numerical results 1 More robust. 2 Better performance on infeasible problems. 3 Dual multipliers are better behaved.

78 Test problems I CUTEst test set I Selected problems with a. at least 100 variables and 100 constraints b. at most 10,000 total variables and constraints I Gives 238 problems I Compare against IPOPT with similar termination criterion ( = 10 6, max iterations = 3000, max time = 1 hour)

79 Comparison on problems where solver outputs are di erent

80 Iteration comparison The x-axis plots: iteration count of solver iteration count of fastest solver

81 Infeasible problems perturbed CUTEst (94 problems) NETLIB (28 problems)

82 Selected problems Table legend: I solver declared problem infeasible I # iterations (time in seconds) name # vars # cons one-phase IPOPT HVYCRASH (8.6 s) 333 (8.2 s) STEERING (1.7 s) 15 (0.2 s) READING9 10,001 5, (7.8 s) 130 (6.7 s) ROBOTARM (64.0 s) ERROR CATENARY (102.2 s) 2581 (909.4 s) WATER SUPPLY 15,935 23, (175.1 s) 1782 (369.9 s) ECON , (563.2 s) out of memory Code and results at

83 Boundedness of dual multipliers on NETLIB LP test set

84 Summary I One-phase IPM for nonconvex optimization I Builds on successful LP implementations [Lustig, 1990, Mehrotra, 1992] I No need for penalty or two-phase method I Circumvents example of Wächter and Biegler [2000] I Comparison against IPOPT on CUTEst test set I Better performance on infeasible problems I More reliable I Better behavior of dual multipliers I Code and test results can be found at

85 Papers 1 Haeser, G., Hinder, O. and Ye, Y., On the behavior of Lagrange multipliers in convex and non-convex infeasible interior point methods. arxiv preprint arxiv: Under review at mathematical programming. 2 Hinder, O. and Ye, Y., A one-phase interior point method for nonconvex optimization. arxiv preprint arxiv: Under review at mathematical programming.

86 Additional slides 1 References 2 Why are local optima are sometimes global? 3 First-order vs. second-order methods 4 AGD for -strongly convex functions 5 Convex until proven guilty numerical results 6 Low tolerance results 7 Runtimes by element

87 References I Erling D Andersen and Yinyu Ye. On a homogeneous algorithm for the monotone complementarity problem. Mathematical Programming, 84(2): , Erling D Andersen, Cornelis Roos, and Tamas Terlaky. On implementing a primal-dual interior-point method for conic quadratic optimization. Mathematical Programming, 95(2): , Richard H Byrd, Jorge Nocedal, and Richard A Waltz. Knitro: An integrated package for nonlinear optimization. In Gianni Di Pillo and M. Roma, editors, Large-scale nonlinear optimization, pages Springer, Lifeng Chen and Donald Goldfarb. Interior-point 2-penalty methods for nonlinear programming with strong global convergence properties. Mathematical Programming, 108(1):1 36, Frank E Curtis. A penalty-interior-point algorithm for nonlinear constrained optimization. Mathematical Programming Computation, pages 1 29, Gabriel Haeser, Oliver Hinder, and Yinyu Ye. On the behavior of Lagrange multipliers in convex and non-convex infeasible interior point methods Oliver Hinder and Yinyu Ye. A one-phase interior point method for non-convex optimization

88 References II Irvin J Lustig. Feasibility issues in a primal-dual interior-point method for linear programming. Mathematical Programming, 49(1): , Kevin A McShane, Clyde L Monma, and David Shanno. An implementation of a primal-dual interior point method for linear programming. ORSA Journal on computing, 1(2):70 83, Sanjay Mehrotra. On the implementation of a primal-dual interior point method. SIAM Journal on optimization, 2(4): , Shinji Mizuno, Michael J Todd, and Yinyu Ye. A surface of analytic centers and primal-dual infeasible-interior-point algorithms for linear programming. Mathematics of Operations Research, 20(1): , Renato DC Monteiro and Ilan Adler. Interior path following primal-dual algorithms. part i: Linear programming. Mathematical programming, 44(1): 27 41, Jos F Sturm. Implementation of interior point methods for mixed semidefinite and second order cone optimization problems. Optimization Methods and Software, 17(6): , 2002.

89 References III Robert J Vanderbei. LOQO user s manual version Optimization Methods and Software, 11(1-4): , Andreas Wächter and Lorenz T Biegler. Failure of global convergence for a class of interior point methods for nonlinear programming. Mathematical Programming, 88(3): , Andreas Wächter and Lorenz T Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming, 106(1):25 57, Yinyu Ye, Michael J Todd, and Shinji Mizuno. An o ( nl)-iteration homogeneous and self-dual linear programming algorithm. Mathematics of Operations Research, 19(1):53 67, 1994.