Inexact Newton Methods and Nonlinear Constrained Optimization
|
|
- Blaise Jordan
- 5 years ago
- Views:
Transcription
1 Inexact Newton Methods and Nonlinear Constrained Optimization Frank E. Curtis EPSRC Symposium Capstone Conference Warwick Mathematics Institute July 2, 2009
2 Outline PDE-Constrained Optimization Newton s method Inexactness Experimental results Conclusion and final remarks
3 Outline PDE-Constrained Optimization Newton s method Inexactness Experimental results Conclusion and final remarks
4 Hyperthermia treatment Regional hyperthermia is a cancer therapy that aims at heating large and deeply seated tumors by means of radio wave adsorption Results in the killing of tumor cells and makes them more susceptible to other accompanying therapies; e.g., chemotherapy
5 Hyperthermia treatment planning Computer modeling can be used to help plan the therapy for each patient, and it opens the door for numerical optimization The goal is to heat the tumor to a target temperature of 43 C while minimizing damage to nearby cells
6 PDE-constrained optimization min f (x) s.t. c E(x) = 0 c I(x) 0 Problem is infinite-dimensional Controls and states: x = (u, y) Solution methods integrate numerical simulation problem structure optimization algorithms
7 Algorithmic frameworks We hear the phrases: Discretize-then-optimize Optimize-then-discretize I prefer: Discretize the optimization problem min f (x) s.t. c(x) = 0 Discretize the optimality conditions min f h(x) s.t. c h (x) = 0 min f (x) s.t. c(x) = 0 [ f + A, λ c ] = 0 [ ] ( f + A, λ )h = 0 c h Discretize the search direction computation
8 Algorithms Nonlinear elimination min u,y f (u, y) s.t. c(u, y) = 0 min u f (u, y(u)) uf + uy T y f = 0 Reduced-space methods d y : toward satisfying the constraints λ : Lagrange multiplier estimates d u : toward optimality Full-space methods H u 0 A T u d u uf + A T 0 H y A T u λ y d y = y f + A T y λ A u A y 0 δ c
9 Outline PDE-Constrained Optimization Newton s method Inexactness Experimental results Conclusion and final remarks
10 Nonlinear equations Newton s method F(x) = 0 F(x k )d k = F(x k ) Judge progress by the merit function φ(x) 1 F(x 2 k) 2 Direction is one of descent since φ(x k ) T d k = F(x k ) T F(x k )d k = F(x k ) 2 < 0 (Note the consistency between the step computation and merit function!)
11 Equality constrained optimization Consider Lagrangian is min f (x) x R n s.t. c(x) = 0 L(x, λ) f (x) + λ T c(x) so the first-order optimality conditions are [ ] f (x) + c(x)λ L(x, λ) = F(x, λ) = 0 c(x)
12 Merit function Simply minimizing [ ] ϕ(x, λ) = 1 F(x, 2 λ) 2 = 1 f (x) + c(x)λ 2 2 c(x) is generally inappropriate for constrained optimization We use the merit function φ(x; π) f (x) + π c(x) where π is a penalty parameter
13 Minimizing a penalty function Consider the penalty function for min (x 1) 2, s.t. x = 0 i.e. φ(x; π) = (x 1) 2 + π x for different values of the penalty parameter π Figure: π = 1 Figure: π = 2
14 Algorithm 0: Newton method for optimization (Assume the problem is sufficiently convex and regular) for k = 0, 1, 2,... Solve the primal-dual (Newton) equations [ ] [ ] [ ] H(xk, λ k ) c(x k ) dk f (xk ) + c(x c(x k ) T = k )λ k 0 c(x k ) δ k Increase π, if necessary, so that Dφ k (d k ; π k ) 0 (e.g., π k λ k + δ k ) Backtrack from α k 1 to satisfy the Armijo condition φ(x k + α k d k ; π k ) φ(x k ; π k ) + ηα k Dφ k (d k ; π k ) Update iterate (x k+1, λ k+1 ) (x k, λ k ) + α k (d k, δ k )
15 Convergence of Algorithm 0 Assumption The sequence {(x k, λ k )} is contained in a convex set Ω over which f, c, and their first derivatives are bounded and Lipschitz continuous. Also, (Regularity) c(x k ) T has full row rank with singular values bounded below by a positive constant (Convexity) u T H(x k, λ k )u µ u 2 for µ > 0 for all u R n satisfying u 0 and c(x k ) T u = 0 Theorem (Han (1977)) The sequence {(x k, λ k )} yields the limit [ ] lim f (xk ) + c(x k )λ k k = 0 c(x k )
16 Outline PDE-Constrained Optimization Newton s method Inexactness Experimental results Conclusion and final remarks
17 Large-scale primal-dual algorithms Computational issues: Large matrices to be stored Large matrices to be factored Algorithmic issues: The problem may be nonconvex The problem may be ill-conditioned Computational/Algorithmic issues: No matrix factorizations makes difficulties more difficult
18 Nonlinear equations Compute F(x k )d k = F(x k ) + r k requiring (Dembo, Eisenstat, Steihaug (1982)) r k κ F(x k ), κ (0, 1) Progress judged by the merit function φ(x) 1 F(x 2 k) 2 Again, note the consistency... φ(x k ) T d k = F(x k ) T F(x k )d k = F(x k ) 2 +F(x k ) T r k (κ 1) F(x k ) 2 < 0
19 Optimization Compute [ ] [ ] [ ] H(xk, λ k ) c(x k ) dk f (xk ) + c(x c(x k ) T = k )λ k + 0 c(x k ) δ k [ ρk r k ] satisfying [ ρk r k ] [ ] κ f (xk ) + c(x k )λ k, κ (0, 1) c(x k ) If κ is not sufficiently small (e.g., 10 3 vs ), then d k may be an ascent direction for our merit function; i.e., Dφ k (d k ; π k ) > 0 for all π k π k 1 Our work begins here... inexact Newton methods for optimization We cover the convex case, nonconvexity, irregularity, inequality constraints
20 Model reductions Define the model of φ(x; π): m(d; π) f (x) + f (x) T d + π( c(x) + c(x) T d ) d k is acceptable if m(d k ; π k ) m(0; π k ) m(d k ; π k ) = f (x k ) T d k + π k ( c(x k ) c(x k ) + c(x k ) T d k ) 0 This ensures Dφ k (d k ; π k ) 0 (and more)
21 Termination test 1 The search direction (d k, δ k ) is acceptable if [ ] [ ] ρk κ f (xk ) + c(x k )λ k, κ (0, 1) c(x k ) r k and if for π k = π k 1 and some σ (0, 1) we have m(d k ; π k ) max{ 1 2 d T k H(x k, λ k )d k, 0} + σπ k max{ c(x k ), r k c(x k ) } }{{} 0 for any d
22 Termination test 2 The search direction (d k, δ k ) is acceptable if ρ k β c(x k ), β > 0 and r k ɛ c(x k ), ɛ (0, 1) Increasing the penalty parameter π then yields m(d k ; π k ) max{ 1 2 d T k H(x k, λ k )d k, 0} + σπ k c(x k ) }{{} 0 for any d
23 Algorithm 1: Inexact Newton for optimization (Byrd, Curtis, Nocedal (2008)) for k = 0, 1, 2,... Iteratively solve [ ] [ ] [ ] H(xk, λ k ) c(x k ) dk f (xk ) + c(x c(x k ) T = k )λ k 0 c(x k ) until termination test 1 or 2 is satisfied If only termination test 2 is satisfied, increase π so { π k max π k 1, f (x k) T d k + max{ 1 d } T 2 k H(x k, λ k )d k, 0} (1 τ)( c(x k ) r k ) Backtrack from α k 1 to satisfy δ k φ(x k + α k d k ; π k ) φ(x k ; π k ) ηα k m(d k ; π k ) Update iterate (x k+1, λ k+1 ) (x k, λ k ) + α k (d k, δ k )
24 Convergence of Algorithm 1 Assumption The sequence {(x k, λ k )} is contained in a convex set Ω over which f, c, and their first derivatives are bounded and Lipschitz continuous. Also, (Regularity) c(x k ) T has full row rank with singular values bounded below by a positive constant (Convexity) u T H(x k, λ k )u µ u 2 for µ > 0 for all u R n satisfying u 0 and c(x k ) T u = 0 Theorem (Byrd, Curtis, Nocedal (2008)) The sequence {(x k, λ k )} yields the limit [ ] lim f (xk ) + c(x k )λ k k = 0 c(x k )
25 Handling nonconvexity and rank deficiency There are two assumptions we aim to drop: (Regularity) c(xk ) T has full row rank with singular values bounded below by a positive constant (Convexity) u T H(x k, λ k )u µ u 2 for µ > 0 for all u R n satisfying u 0 and c(x k ) T u = 0 e.g., the problem is not regular if it is infeasible, and it is not convex if there are maximizers and/or saddle points Without them, Algorithm 1 may stall or may not be well-defined
26 No factorizations means no clue We might not store or factor [ H(xk, λ k ) ] c(x k ) c(x k ) T 0 so we might not know if the problem is nonconvex or ill-conditioned Common practice is to perturb the matrix to be [ ] H(xk, λ k ) + ξ 1I c(x k ) c(x k ) T ξ 2I where ξ 1 convexifies the model and ξ 2 regularizes the constraints Poor choices of ξ 1 and ξ 2 can have terrible consequences in the algorithm
27 Our approach for global convergence Decompose the direction d k into a normal component (toward the constraints) and a tangential component (toward optimality) We impose a specific type of trust region constraint on the v k step in case the constraint Jacobian is (near) rank deficient
28 Handling nonconvexity In computation of d k = v k + u k, convexify the Hessian as in [ H(xk, λ k ) + ξ 1I ] c(x k ) c(x k ) T 0 by monitoring iterates Hessian modification strategy: Increase ξ 1 whenever u k 2 > ψ v k 2, ψ > ut k (H(x k, λ k ) + ξ 1I )u k < θ u k 2, θ > 0
29 Algorithm 2: Inexact Newton (Regularized) (Curtis, Nocedal, Wächter (2009)) for k = 0, 1, 2,... Approximately solve min 1 2 c(x k) + c(x k ) T v 2, s.t. v ω ( c(x k ))c(x k ) to compute v k satisfying Cauchy decrease Iteratively solve [ ] [ ] [ ] H(xk, λ k ) + ξ 1 I c(x k ) dk f (xk ) + c(x c(x k ) T = k )λ k 0 δ k c(x k ) T v k until termination test 1 or 2 is satisfied, increasing ξ 1 as described If only termination test 2 is satisfied, increase π so { π k max π k 1, f (x k) T d k + max{ 1 2 ut k (H(x } k, λ k ) + ξ 1 I )u k, θ u k 2 } (1 τ)( c(x k ) c(x k ) + c(x k ) T d k ) Backtrack from α k 1 to satisfy φ(x k + α k d k ; π k ) φ(x k ; π k ) ηα k m(d k ; π k ) Update iterate (x k+1, λ k+1 ) (x k, λ k ) + α k (d k, δ k )
30 Convergence of Algorithm 2 Assumption The sequence {(x k, λ k )} is contained in a convex set Ω over which f, c, and their first derivatives are bounded and Lipschitz continuous Theorem (Curtis, Nocedal, Wächter (2009)) If all limit points of { c(x k ) T } have full row rank, then the sequence {(x k, λ k )} yields the limit [ ] lim f (xk ) + c(x k )λ k k = 0. c(x k ) Otherwise, and if {π k } is bounded, then lim ( c(x k))c(x k ) = 0 k lim f (x k) + c(x k )λ k = 0 k
31 Handling inequalities Interior point methods are attractive for large applications Line-search interior point methods that enforce c(x k ) + c(x k ) T d k = 0 may fail to converge globally (Wächter, Biegler (2000)) Fortunately, the trust region subproblem we use to regularize the constraints also saves us from this type of failure!
32 Algorithm 2 (Interior-point version) Apply Algorithm 2 to the logarithmic-barrier subproblem for µ 0 Define min f (x) µ q ln s i, s.t. c E (x) = 0, c I (x) s = 0 i=1 H(x k, λ E,k, λ I,k ) 0 c E (x k ) c I (x k ) d x k 0 µi 0 S k c E (x k ) T d s k δ E,k c I (x k ) T S k 0 0 δ I,k so that the iterate update has [ ] [ ] [ ] xk+1 xk d x + α k s k+1 s k k S k dk s Incorporate a fraction-to-the-boundary rule in the line search and a slack reset in the algorithm to maintain s max{0, c I (x)}
33 Convergence of Algorithm 2 (Interior-point) Assumption The sequence {(x k, λ E,k, λ I,k )} is contained in a convex set Ω over which f, c E, c I, and their first derivatives are bounded and Lipschitz continuous Theorem (Curtis, Schenk, Wächter (2009)) For a given µ, Algorithm 2 yields the same limits as in the equality constrained case If Algorithm 2 yields a sufficiently accurate solution to the barrier subproblem for each {µ j } 0 and if the linear independence constraint qualification (LICQ) holds at a limit point x of {x j }, then there exist Lagrange multipliers λ such that the first-order optimality conditions of the nonlinear program are satisfied
34 Outline PDE-Constrained Optimization Newton s method Inexactness Experimental results Conclusion and final remarks
35 Implementation details Incorporated in IPOPT software package (Wächter) inexact algorithm yes Linear systems solved with PARDISO (Schenk) SQMR (Freund (1994)) Preconditioning in PARDISO incomplete multilevel factorization with inverse-based pivoting stabilized by symmetric-weighted matchings Optimality tolerance: 1e-8
36 CUTEr and COPS collections 745 problems written in AMPL 645 solved successfully 42 real failures Robustness between 87%-94% Original IPOPT: 93%
37 Helmholtz N n p q # iter CPU sec (per iter) (21.833) (149.66) (2729.1)
38 Helmholtz Not taking nonconvexity into account:
39 Boundary control min 1 2 Ω (y(x) yt(x))2 dx s.t. (e y(x) y(x)) = 20 in Ω y(x) = u(x) on Ω 2.5 u(x) 3.5 on Ω where y t(x) = x 1 (x 1 1)x 2 (x 2 1) sin(2πx 3 ) N n p q # iter CPU sec (per iter) (0.2165) (7.9731) (380.88) Original IPOPT with N = 32 requires 238 seconds per iteration
40 Hyperthermia Treatment Planning min 1 2 Ω (y(x) yt(x))2 dx s.t. y(x) 10(y(x) 37) = u M(x)u in Ω 37.0 y(x) 37.5 on Ω 42.0 y(x) 44.0 in Ω 0 where u j = a j e iφ j, M jk (x) =< E j (x), E k (x) >, E j = sin(jx 1 x 2 x 3 π) N n p q # iter CPU sec (per iter) (0.3367) (59.920) Original IPOPT with N = 32 requires 408 seconds per iteration
41 Groundwater modeling min 1 2 Ω (y(x) yt(x))2 dx α Ω [β(u(x) ut(x))2 + (u(x) u t(x)) 2 ]dx s.t. (e u(x) y i (x)) = q i (x) in Ω, i = 1,..., 6 y i (x) n = 0 on Ω y i (x)dx = 0, i = 1,..., 6 Ω 1 u(x) 2 in Ω where q i = 100 sin(2πx 1 ) sin(2πx 2 ) sin(2πx 3 ) N n p q # iter CPU sec (per iter) ( ) ( ) ( ) Original IPOPT with N = 32 requires approx. 20 hours for the first iteration
42 Outline PDE-Constrained Optimization Newton s method Inexactness Experimental results Conclusion and final remarks
43 Conclusion and final remarks PDE-Constrained optimization is an active and exciting area Inexact Newton method with theoretical foundation Convergence guarantees are as good as exact methods, sometimes better Numerical experiments are promising so far, and more to come
An Inexact Newton Method for Optimization
New York University Brown Applied Mathematics Seminar, February 10, 2009 Brief biography New York State College of William and Mary (B.S.) Northwestern University (M.S. & Ph.D.) Courant Institute (Postdoc)
More informationPDE-Constrained and Nonsmooth Optimization
Frank E. Curtis October 1, 2009 Outline PDE-Constrained Optimization Introduction Newton s method Inexactness Results Summary and future work Nonsmooth Optimization Sequential quadratic programming (SQP)
More informationAn Inexact Newton Method for Nonlinear Constrained Optimization
An Inexact Newton Method for Nonlinear Constrained Optimization Frank E. Curtis Numerical Analysis Seminar, January 23, 2009 Outline Motivation and background Algorithm development and theoretical results
More informationLarge-Scale Nonlinear Optimization with Inexact Step Computations
Large-Scale Nonlinear Optimization with Inexact Step Computations Andreas Wächter IBM T.J. Watson Research Center Yorktown Heights, New York andreasw@us.ibm.com IPAM Workshop on Numerical Methods for Continuous
More informationNumerical Methods for PDE-Constrained Optimization
Numerical Methods for PDE-Constrained Optimization Richard H. Byrd 1 Frank E. Curtis 2 Jorge Nocedal 2 1 University of Colorado at Boulder 2 Northwestern University Courant Institute of Mathematical Sciences,
More informationRecent Adaptive Methods for Nonlinear Optimization
Recent Adaptive Methods for Nonlinear Optimization Frank E. Curtis, Lehigh University involving joint work with James V. Burke (U. of Washington), Richard H. Byrd (U. of Colorado), Nicholas I. M. Gould
More informationInfeasibility Detection and an Inexact Active-Set Method for Large-Scale Nonlinear Optimization
Infeasibility Detection and an Inexact Active-Set Method for Large-Scale Nonlinear Optimization Frank E. Curtis, Lehigh University involving joint work with James V. Burke, University of Washington Daniel
More informationAn Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization
An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization Frank E. Curtis, Lehigh University involving joint work with Travis Johnson, Northwestern University Daniel P. Robinson, Johns
More information1. Introduction. We consider nonlinear optimization problems of the form. f(x) ce (x) = 0 c I (x) 0,
AN INTERIOR-POINT ALGORITHM FOR LARGE-SCALE NONLINEAR OPTIMIZATION WITH INEXACT STEP COMPUTATIONS FRANK E. CURTIS, OLAF SCHENK, AND ANDREAS WÄCHTER Abstract. We present a line-search algorithm for large-scale
More informationOn the Implementation of an Interior-Point Algorithm for Nonlinear Optimization with Inexact Step Computations
On the Implementation of an Interior-Point Algorithm for Nonlinear Optimization with Inexact Step Computations Frank E. Curtis Lehigh University Johannes Huber University of Basel Olaf Schenk University
More informationAn Inexact Newton Method for Nonconvex Equality Constrained Optimization
Noname manuscript No. (will be inserted by the editor) An Inexact Newton Method for Nonconvex Equality Constrained Optimization Richard H. Byrd Frank E. Curtis Jorge Nocedal Received: / Accepted: Abstract
More information1. Introduction. In this paper we discuss an algorithm for equality constrained optimization problems of the form. f(x) s.t.
AN INEXACT SQP METHOD FOR EQUALITY CONSTRAINED OPTIMIZATION RICHARD H. BYRD, FRANK E. CURTIS, AND JORGE NOCEDAL Abstract. We present an algorithm for large-scale equality constrained optimization. The
More informationAlgorithms for Constrained Optimization
1 / 42 Algorithms for Constrained Optimization ME598/494 Lecture Max Yi Ren Department of Mechanical Engineering, Arizona State University April 19, 2015 2 / 42 Outline 1. Convergence 2. Sequential quadratic
More informationPenalty and Barrier Methods General classical constrained minimization problem minimize f(x) subject to g(x) 0 h(x) =0 Penalty methods are motivated by the desire to use unconstrained optimization techniques
More information5 Handling Constraints
5 Handling Constraints Engineering design optimization problems are very rarely unconstrained. Moreover, the constraints that appear in these problems are typically nonlinear. This motivates our interest
More informationConstrained Nonlinear Optimization Algorithms
Department of Industrial Engineering and Management Sciences Northwestern University waechter@iems.northwestern.edu Institute for Mathematics and its Applications University of Minnesota August 4, 2016
More informationMS&E 318 (CME 338) Large-Scale Numerical Optimization
Stanford University, Management Science & Engineering (and ICME) MS&E 318 (CME 338) Large-Scale Numerical Optimization 1 Origins Instructor: Michael Saunders Spring 2015 Notes 9: Augmented Lagrangian Methods
More informationA GLOBALLY CONVERGENT STABILIZED SQP METHOD
A GLOBALLY CONVERGENT STABILIZED SQP METHOD Philip E. Gill Daniel P. Robinson July 6, 2013 Abstract Sequential quadratic programming SQP methods are a popular class of methods for nonlinearly constrained
More informationPart 5: Penalty and augmented Lagrangian methods for equality constrained optimization. Nick Gould (RAL)
Part 5: Penalty and augmented Lagrangian methods for equality constrained optimization Nick Gould (RAL) x IR n f(x) subject to c(x) = Part C course on continuoue optimization CONSTRAINED MINIMIZATION x
More informationA Penalty-Interior-Point Algorithm for Nonlinear Constrained Optimization
Noname manuscript No. (will be inserted by the editor) A Penalty-Interior-Point Algorithm for Nonlinear Constrained Optimization Frank E. Curtis April 26, 2011 Abstract Penalty and interior-point methods
More informationA SHIFTED PRIMAL-DUAL PENALTY-BARRIER METHOD FOR NONLINEAR OPTIMIZATION
A SHIFTED PRIMAL-DUAL PENALTY-BARRIER METHOD FOR NONLINEAR OPTIMIZATION Philip E. Gill Vyacheslav Kungurtsev Daniel P. Robinson UCSD Center for Computational Mathematics Technical Report CCoM-19-3 March
More informationINTERIOR-POINT METHODS FOR NONCONVEX NONLINEAR PROGRAMMING: CONVERGENCE ANALYSIS AND COMPUTATIONAL PERFORMANCE
INTERIOR-POINT METHODS FOR NONCONVEX NONLINEAR PROGRAMMING: CONVERGENCE ANALYSIS AND COMPUTATIONAL PERFORMANCE HANDE Y. BENSON, ARUN SEN, AND DAVID F. SHANNO Abstract. In this paper, we present global
More informationLecture 11 and 12: Penalty methods and augmented Lagrangian methods for nonlinear programming
Lecture 11 and 12: Penalty methods and augmented Lagrangian methods for nonlinear programming Coralia Cartis, Mathematical Institute, University of Oxford C6.2/B2: Continuous Optimization Lecture 11 and
More informationTrust-Region SQP Methods with Inexact Linear System Solves for Large-Scale Optimization
Trust-Region SQP Methods with Inexact Linear System Solves for Large-Scale Optimization Denis Ridzal Department of Computational and Applied Mathematics Rice University, Houston, Texas dridzal@caam.rice.edu
More informationA Note on the Implementation of an Interior-Point Algorithm for Nonlinear Optimization with Inexact Step Computations
Noname manuscript No. (will be inserted by the editor) A Note on the Implementation of an Interior-Point Algorithm for Nonlinear Optimization with Inexact Step Computations Frank E. Curtis Johannes Huber
More informationA SHIFTED PRIMAL-DUAL INTERIOR METHOD FOR NONLINEAR OPTIMIZATION
A SHIFTED RIMAL-DUAL INTERIOR METHOD FOR NONLINEAR OTIMIZATION hilip E. Gill Vyacheslav Kungurtsev Daniel. Robinson UCSD Center for Computational Mathematics Technical Report CCoM-18-1 February 1, 2018
More informationWhat s New in Active-Set Methods for Nonlinear Optimization?
What s New in Active-Set Methods for Nonlinear Optimization? Philip E. Gill Advances in Numerical Computation, Manchester University, July 5, 2011 A Workshop in Honor of Sven Hammarling UCSD Center for
More informationPenalty and Barrier Methods. So we again build on our unconstrained algorithms, but in a different way.
AMSC 607 / CMSC 878o Advanced Numerical Optimization Fall 2008 UNIT 3: Constrained Optimization PART 3: Penalty and Barrier Methods Dianne P. O Leary c 2008 Reference: N&S Chapter 16 Penalty and Barrier
More informationA GLOBALLY CONVERGENT STABILIZED SQP METHOD: SUPERLINEAR CONVERGENCE
A GLOBALLY CONVERGENT STABILIZED SQP METHOD: SUPERLINEAR CONVERGENCE Philip E. Gill Vyacheslav Kungurtsev Daniel P. Robinson UCSD Center for Computational Mathematics Technical Report CCoM-14-1 June 30,
More informationA STABILIZED SQP METHOD: SUPERLINEAR CONVERGENCE
A STABILIZED SQP METHOD: SUPERLINEAR CONVERGENCE Philip E. Gill Vyacheslav Kungurtsev Daniel P. Robinson UCSD Center for Computational Mathematics Technical Report CCoM-14-1 June 30, 2014 Abstract Regularized
More informationHot-Starting NLP Solvers
Hot-Starting NLP Solvers Andreas Wächter Department of Industrial Engineering and Management Sciences Northwestern University waechter@iems.northwestern.edu 204 Mixed Integer Programming Workshop Ohio
More informationA Primal-Dual Augmented Lagrangian Penalty-Interior-Point Filter Line Search Algorithm
Journal name manuscript No. (will be inserted by the editor) A Primal-Dual Augmented Lagrangian Penalty-Interior-Point Filter Line Search Algorithm Rene Kuhlmann Christof Büsens Received: date / Accepted:
More informationCONVERGENCE ANALYSIS OF AN INTERIOR-POINT METHOD FOR NONCONVEX NONLINEAR PROGRAMMING
CONVERGENCE ANALYSIS OF AN INTERIOR-POINT METHOD FOR NONCONVEX NONLINEAR PROGRAMMING HANDE Y. BENSON, ARUN SEN, AND DAVID F. SHANNO Abstract. In this paper, we present global and local convergence results
More informationAlgorithms for constrained local optimization
Algorithms for constrained local optimization Fabio Schoen 2008 http://gol.dsi.unifi.it/users/schoen Algorithms for constrained local optimization p. Feasible direction methods Algorithms for constrained
More informationA PRIMAL-DUAL TRUST REGION ALGORITHM FOR NONLINEAR OPTIMIZATION
Optimization Technical Report 02-09, October 2002, UW-Madison Computer Sciences Department. E. Michael Gertz 1 Philip E. Gill 2 A PRIMAL-DUAL TRUST REGION ALGORITHM FOR NONLINEAR OPTIMIZATION 7 October
More information10 Numerical methods for constrained problems
10 Numerical methods for constrained problems min s.t. f(x) h(x) = 0 (l), g(x) 0 (m), x X The algorithms can be roughly divided the following way: ˆ primal methods: find descent direction keeping inside
More informationA Penalty-Interior-Point Algorithm for Nonlinear Constrained Optimization
Noname manuscript No. (will be inserted by the editor) A Penalty-Interior-Point Algorithm for Nonlinear Constrained Optimization Fran E. Curtis February, 22 Abstract Penalty and interior-point methods
More informationFeasible Interior Methods Using Slacks for Nonlinear Optimization
Feasible Interior Methods Using Slacks for Nonlinear Optimization Richard H. Byrd Jorge Nocedal Richard A. Waltz February 28, 2005 Abstract A slack-based feasible interior point method is described which
More informationSurvey of NLP Algorithms. L. T. Biegler Chemical Engineering Department Carnegie Mellon University Pittsburgh, PA
Survey of NLP Algorithms L. T. Biegler Chemical Engineering Department Carnegie Mellon University Pittsburgh, PA NLP Algorithms - Outline Problem and Goals KKT Conditions and Variable Classification Handling
More informationA DECOMPOSITION PROCEDURE BASED ON APPROXIMATE NEWTON DIRECTIONS
Working Paper 01 09 Departamento de Estadística y Econometría Statistics and Econometrics Series 06 Universidad Carlos III de Madrid January 2001 Calle Madrid, 126 28903 Getafe (Spain) Fax (34) 91 624
More informationA Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization with O(ɛ 3/2 ) Complexity
A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization with O(ɛ 3/2 ) Complexity Mohammadreza Samadi, Lehigh University joint work with Frank E. Curtis (stand-in presenter), Lehigh University
More informationDerivative-Based Numerical Method for Penalty-Barrier Nonlinear Programming
Derivative-Based Numerical Method for Penalty-Barrier Nonlinear Programming Martin Peter Neuenhofen October 26, 2018 Abstract We present an NLP solver for nonlinear optimization with quadratic penalty
More informationA New Penalty-SQP Method
Background and Motivation Illustration of Numerical Results Final Remarks Frank E. Curtis Informs Annual Meeting, October 2008 Background and Motivation Illustration of Numerical Results Final Remarks
More informationSequential Quadratic Programming Method for Nonlinear Second-Order Cone Programming Problems. Hirokazu KATO
Sequential Quadratic Programming Method for Nonlinear Second-Order Cone Programming Problems Guidance Professor Masao FUKUSHIMA Hirokazu KATO 2004 Graduate Course in Department of Applied Mathematics and
More informationOn the use of piecewise linear models in nonlinear programming
Math. Program., Ser. A (2013) 137:289 324 DOI 10.1007/s10107-011-0492-9 FULL LENGTH PAPER On the use of piecewise linear models in nonlinear programming Richard H. Byrd Jorge Nocedal Richard A. Waltz Yuchen
More informationA globally and quadratically convergent primal dual augmented Lagrangian algorithm for equality constrained optimization
Optimization Methods and Software ISSN: 1055-6788 (Print) 1029-4937 (Online) Journal homepage: http://www.tandfonline.com/loi/goms20 A globally and quadratically convergent primal dual augmented Lagrangian
More informationAn Interior Algorithm for Nonlinear Optimization That Combines Line Search and Trust Region Steps
An Interior Algorithm for Nonlinear Optimization That Combines Line Search and Trust Region Steps R.A. Waltz J.L. Morales J. Nocedal D. Orban September 8, 2004 Abstract An interior-point method for nonlinear
More informationPart 4: Active-set methods for linearly constrained optimization. Nick Gould (RAL)
Part 4: Active-set methods for linearly constrained optimization Nick Gould RAL fx subject to Ax b Part C course on continuoue optimization LINEARLY CONSTRAINED MINIMIZATION fx subject to Ax { } b where
More informationREGULARIZED SEQUENTIAL QUADRATIC PROGRAMMING METHODS
REGULARIZED SEQUENTIAL QUADRATIC PROGRAMMING METHODS Philip E. Gill Daniel P. Robinson UCSD Department of Mathematics Technical Report NA-11-02 October 2011 Abstract We present the formulation and analysis
More informationAn Active Set Strategy for Solving Optimization Problems with up to 200,000,000 Nonlinear Constraints
An Active Set Strategy for Solving Optimization Problems with up to 200,000,000 Nonlinear Constraints Klaus Schittkowski Department of Computer Science, University of Bayreuth 95440 Bayreuth, Germany e-mail:
More informationA Line search Multigrid Method for Large-Scale Nonlinear Optimization
A Line search Multigrid Method for Large-Scale Nonlinear Optimization Zaiwen Wen Donald Goldfarb Department of Industrial Engineering and Operations Research Columbia University 2008 Siam Conference on
More informationA STABILIZED SQP METHOD: GLOBAL CONVERGENCE
A STABILIZED SQP METHOD: GLOBAL CONVERGENCE Philip E. Gill Vyacheslav Kungurtsev Daniel P. Robinson UCSD Center for Computational Mathematics Technical Report CCoM-13-4 Revised July 18, 2014, June 23,
More information2.3 Linear Programming
2.3 Linear Programming Linear Programming (LP) is the term used to define a wide range of optimization problems in which the objective function is linear in the unknown variables and the constraints are
More informationImplementation of an Interior Point Multidimensional Filter Line Search Method for Constrained Optimization
Proceedings of the 5th WSEAS Int. Conf. on System Science and Simulation in Engineering, Tenerife, Canary Islands, Spain, December 16-18, 2006 391 Implementation of an Interior Point Multidimensional Filter
More informationminimize x subject to (x 2)(x 4) u,
Math 6366/6367: Optimization and Variational Methods Sample Preliminary Exam Questions 1. Suppose that f : [, L] R is a C 2 -function with f () on (, L) and that you have explicit formulae for
More information12. Interior-point methods
12. Interior-point methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity
More informationLecture 13: Constrained optimization
2010-12-03 Basic ideas A nonlinearly constrained problem must somehow be converted relaxed into a problem which we can solve (a linear/quadratic or unconstrained problem) We solve a sequence of such problems
More informationSF2822 Applied Nonlinear Optimization. Preparatory question. Lecture 10: Interior methods. Anders Forsgren. 1. Try to solve theory question 7.
SF2822 Applied Nonlinear Optimization Lecture 10: Interior methods Anders Forsgren SF2822 Applied Nonlinear Optimization, KTH 1 / 24 Lecture 10, 2017/2018 Preparatory question 1. Try to solve theory question
More information1 Computing with constraints
Notes for 2017-04-26 1 Computing with constraints Recall that our basic problem is minimize φ(x) s.t. x Ω where the feasible set Ω is defined by equality and inequality conditions Ω = {x R n : c i (x)
More informationOptimisation in Higher Dimensions
CHAPTER 6 Optimisation in Higher Dimensions Beyond optimisation in 1D, we will study two directions. First, the equivalent in nth dimension, x R n such that f(x ) f(x) for all x R n. Second, constrained
More informationLINEAR AND NONLINEAR PROGRAMMING
LINEAR AND NONLINEAR PROGRAMMING Stephen G. Nash and Ariela Sofer George Mason University The McGraw-Hill Companies, Inc. New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico
More informationarxiv: v1 [math.na] 8 Jun 2018
arxiv:1806.03347v1 [math.na] 8 Jun 2018 Interior Point Method with Modified Augmented Lagrangian for Penalty-Barrier Nonlinear Programming Martin Neuenhofen June 12, 2018 Abstract We present a numerical
More informationLecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem
Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem Michael Patriksson 0-0 The Relaxation Theorem 1 Problem: find f := infimum f(x), x subject to x S, (1a) (1b) where f : R n R
More informationA null-space primal-dual interior-point algorithm for nonlinear optimization with nice convergence properties
A null-space primal-dual interior-point algorithm for nonlinear optimization with nice convergence properties Xinwei Liu and Yaxiang Yuan Abstract. We present a null-space primal-dual interior-point algorithm
More informationA Trust-Funnel Algorithm for Nonlinear Programming
for Nonlinear Programming Daniel P. Robinson Johns Hopins University Department of Applied Mathematics and Statistics Collaborators: Fran E. Curtis (Lehigh University) Nic I. M. Gould (Rutherford Appleton
More informationWritten Examination
Division of Scientific Computing Department of Information Technology Uppsala University Optimization Written Examination 202-2-20 Time: 4:00-9:00 Allowed Tools: Pocket Calculator, one A4 paper with notes
More informationAN INTERIOR-POINT METHOD FOR NONLINEAR OPTIMIZATION PROBLEMS WITH LOCATABLE AND SEPARABLE NONSMOOTHNESS
AN INTERIOR-POINT METHOD FOR NONLINEAR OPTIMIZATION PROBLEMS WITH LOCATABLE AND SEPARABLE NONSMOOTHNESS MARTIN SCHMIDT Abstract. Many real-world optimization models comse nonconvex and nonlinear as well
More informationApplications of Linear Programming
Applications of Linear Programming lecturer: András London University of Szeged Institute of Informatics Department of Computational Optimization Lecture 9 Non-linear programming In case of LP, the goal
More information5 Quasi-Newton Methods
Unconstrained Convex Optimization 26 5 Quasi-Newton Methods If the Hessian is unavailable... Notation: H = Hessian matrix. B is the approximation of H. C is the approximation of H 1. Problem: Solve min
More informationarxiv: v2 [math.oc] 11 Jan 2018
A one-phase interior point method for nonconvex optimization Oliver Hinder, Yinyu Ye January 12, 2018 arxiv:1801.03072v2 [math.oc] 11 Jan 2018 Abstract The work of Wächter and Biegler [40] suggests that
More informationMS&E 318 (CME 338) Large-Scale Numerical Optimization
Stanford University, Management Science & Engineering (and ICME) MS&E 318 (CME 338) Large-Scale Numerical Optimization Instructor: Michael Saunders Spring 2015 Notes 11: NPSOL and SNOPT SQP Methods 1 Overview
More informationQuiz Discussion. IE417: Nonlinear Programming: Lecture 12. Motivation. Why do we care? Jeff Linderoth. 16th March 2006
Quiz Discussion IE417: Nonlinear Programming: Lecture 12 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 16th March 2006 Motivation Why do we care? We are interested in
More informationLagrangian Duality Theory
Lagrangian Duality Theory Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapter 14.1-4 1 Recall Primal and Dual
More informationConvex Optimization. Newton s method. ENSAE: Optimisation 1/44
Convex Optimization Newton s method ENSAE: Optimisation 1/44 Unconstrained minimization minimize f(x) f convex, twice continuously differentiable (hence dom f open) we assume optimal value p = inf x f(x)
More informationAN AUGMENTED LAGRANGIAN AFFINE SCALING METHOD FOR NONLINEAR PROGRAMMING
AN AUGMENTED LAGRANGIAN AFFINE SCALING METHOD FOR NONLINEAR PROGRAMMING XIAO WANG AND HONGCHAO ZHANG Abstract. In this paper, we propose an Augmented Lagrangian Affine Scaling (ALAS) algorithm for general
More informationBindel, Spring 2017 Numerical Analysis (CS 4220) Notes for So far, we have considered unconstrained optimization problems.
Consider constraints Notes for 2017-04-24 So far, we have considered unconstrained optimization problems. The constrained problem is minimize φ(x) s.t. x Ω where Ω R n. We usually define x in terms of
More informationConvergence analysis of a primal-dual interior-point method for nonlinear programming
Convergence analysis of a primal-dual interior-point method for nonlinear programming Igor Griva David F. Shanno Robert J. Vanderbei August 19, 2005 Abstract We analyze a primal-dual interior-point method
More informationA note on the implementation of an interior-point algorithm for nonlinear optimization with inexact step computations
Math. Program., Ser. B (2012) 136:209 227 DOI 10.1007/s10107-012-0557-4 FULL LENGTH PAPER A note on the implementation of an interior-point algorithm for nonlinear optimization with inexact step computations
More informationSome new facts about sequential quadratic programming methods employing second derivatives
To appear in Optimization Methods and Software Vol. 00, No. 00, Month 20XX, 1 24 Some new facts about sequential quadratic programming methods employing second derivatives A.F. Izmailov a and M.V. Solodov
More informationOptimization. Escuela de Ingeniería Informática de Oviedo. (Dpto. de Matemáticas-UniOvi) Numerical Computation Optimization 1 / 30
Optimization Escuela de Ingeniería Informática de Oviedo (Dpto. de Matemáticas-UniOvi) Numerical Computation Optimization 1 / 30 Unconstrained optimization Outline 1 Unconstrained optimization 2 Constrained
More informationLectures 9 and 10: Constrained optimization problems and their optimality conditions
Lectures 9 and 10: Constrained optimization problems and their optimality conditions Coralia Cartis, Mathematical Institute, University of Oxford C6.2/B2: Continuous Optimization Lectures 9 and 10: Constrained
More information8 Numerical methods for unconstrained problems
8 Numerical methods for unconstrained problems Optimization is one of the important fields in numerical computation, beside solving differential equations and linear systems. We can see that these fields
More informationDetermination of Feasible Directions by Successive Quadratic Programming and Zoutendijk Algorithms: A Comparative Study
International Journal of Mathematics And Its Applications Vol.2 No.4 (2014), pp.47-56. ISSN: 2347-1557(online) Determination of Feasible Directions by Successive Quadratic Programming and Zoutendijk Algorithms:
More informationPrimal-dual Subgradient Method for Convex Problems with Functional Constraints
Primal-dual Subgradient Method for Convex Problems with Functional Constraints Yurii Nesterov, CORE/INMA (UCL) Workshop on embedded optimization EMBOPT2014 September 9, 2014 (Lucca) Yu. Nesterov Primal-dual
More informationComplexity analysis of second-order algorithms based on line search for smooth nonconvex optimization
Complexity analysis of second-order algorithms based on line search for smooth nonconvex optimization Clément Royer - University of Wisconsin-Madison Joint work with Stephen J. Wright MOPTA, Bethlehem,
More informationA Trust-region-based Sequential Quadratic Programming Algorithm
Downloaded from orbit.dtu.dk on: Oct 19, 2018 A Trust-region-based Sequential Quadratic Programming Algorithm Henriksen, Lars Christian; Poulsen, Niels Kjølstad Publication date: 2010 Document Version
More informationPOWER SYSTEMS in general are currently operating
TO APPEAR IN IEEE TRANSACTIONS ON POWER SYSTEMS 1 Robust Optimal Power Flow Solution Using Trust Region and Interior-Point Methods Andréa A. Sousa, Geraldo L. Torres, Member IEEE, Claudio A. Cañizares,
More informationOn the complexity of an Inexact Restoration method for constrained optimization
On the complexity of an Inexact Restoration method for constrained optimization L. F. Bueno J. M. Martínez September 18, 2018 Abstract Recent papers indicate that some algorithms for constrained optimization
More information4TE3/6TE3. Algorithms for. Continuous Optimization
4TE3/6TE3 Algorithms for Continuous Optimization (Algorithms for Constrained Nonlinear Optimization Problems) Tamás TERLAKY Computing and Software McMaster University Hamilton, November 2005 terlaky@mcmaster.ca
More informationInfeasibility Detection in Nonlinear Optimization
Infeasibility Detection in Nonlinear Optimization Frank E. Curtis, Lehigh University Hao Wang, Lehigh University SIAM Conference on Optimization 2011 May 16, 2011 (See also Infeasibility Detection in Nonlinear
More informationA Primal-Dual Interior-Point Method for Nonlinear Programming with Strong Global and Local Convergence Properties
A Primal-Dual Interior-Point Method for Nonlinear Programming with Strong Global and Local Convergence Properties André L. Tits Andreas Wächter Sasan Bahtiari Thomas J. Urban Craig T. Lawrence ISR Technical
More informationx 2. x 1
Feasibility Control in Nonlinear Optimization y M. Marazzi z Jorge Nocedal x March 9, 2000 Report OTC 2000/04 Optimization Technology Center Abstract We analyze the properties that optimization algorithms
More informationBarrier Method. Javier Peña Convex Optimization /36-725
Barrier Method Javier Peña Convex Optimization 10-725/36-725 1 Last time: Newton s method For root-finding F (x) = 0 x + = x F (x) 1 F (x) For optimization x f(x) x + = x 2 f(x) 1 f(x) Assume f strongly
More informationE5295/5B5749 Convex optimization with engineering applications. Lecture 8. Smooth convex unconstrained and equality-constrained minimization
E5295/5B5749 Convex optimization with engineering applications Lecture 8 Smooth convex unconstrained and equality-constrained minimization A. Forsgren, KTH 1 Lecture 8 Convex optimization 2006/2007 Unconstrained
More informationarxiv: v1 [math.oc] 20 Aug 2014
Adaptive Augmented Lagrangian Methods: Algorithms and Practical Numerical Experience Detailed Version Frank E. Curtis, Nicholas I. M. Gould, Hao Jiang, and Daniel P. Robinson arxiv:1408.4500v1 [math.oc]
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 6 Optimization Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 6 Optimization Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted
More informationInterior Methods for Mathematical Programs with Complementarity Constraints
Interior Methods for Mathematical Programs with Complementarity Constraints Sven Leyffer, Gabriel López-Calva and Jorge Nocedal July 14, 25 Abstract This paper studies theoretical and practical properties
More informationOn the Local Quadratic Convergence of the Primal-Dual Augmented Lagrangian Method
Optimization Methods and Software Vol. 00, No. 00, Month 200x, 1 11 On the Local Quadratic Convergence of the Primal-Dual Augmented Lagrangian Method ROMAN A. POLYAK Department of SEOR and Mathematical
More informationSolving Dual Problems
Lecture 20 Solving Dual Problems We consider a constrained problem where, in addition to the constraint set X, there are also inequality and linear equality constraints. Specifically the minimization problem
More information