Nonlinear Programming Computation, Applications, Software Continuous Optimization: Recent Developments and Applications Stephen Wright Department of Computer Sciences University of Wisconsin-Madison SIAM, Portland, July, 2004 Stephen Wright Continuous Optimization: Recent Developments
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Continuous Optimization: Recent Developments and Applicationspages.cs.wisc.edu/~swright/talks/siam-annual-jul04.pdf · Computation, Applications, Software Optimization on the Grid
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Since NLP algorithms work with approximations based on thealgebraic representation of the feasible set, need conditions toensure that such approximations capture the essential geometry ofthe set: Constraint Qualifications.
Define A∗ def= i = 1, 2, . . . ,m | ci (x
∗) = 0.
I Linear Independence (LICQ): The gradients ∇ci (x∗), i ∈ A∗
are linearly independent.
I Mangasarian-Fromovitz (MFCQ): There is a direction d suchthat ∇ci (x
∗)Td > 0 for all i ∈ A∗.
Stephen Wright Continuous Optimization: Recent Developments
Optimality conditions can be expressed in terms of the Lagrangianfunction, which is a linear combination of objectives andconstraints, with Lagrange multipliers as coefficients:
L(x , λ) = f (x)− λT c(x).
Can write KKT conditions as
∇xL(x∗, λ∗) = 0,
0 ≤ c(x∗) ⊥ λ∗ ≥ 0.
Stephen Wright Continuous Optimization: Recent Developments
Many NLP algorithms use a merit function to gauge progress andto decide whether to accept a candidate step. Usually a weightedaverage of objective value and constraint violation, e.g.
P(x ; ν) = f (x) + νr(c(x)), where
r(c(x))def=
m∑i=1
max(0,−ci (x)) = ‖max(0,−c(x))‖1.
Choosing the penalty parameter ν may be tricky.Too small: P(x ; ν) unbounded below;Too large: Slow progress along boundary of Ω.
Seek a “minimalist” approach that doesn’t require choice of aparameter, and rejects good steps less often than does P.
Stephen Wright Continuous Optimization: Recent Developments
Following original proposal by Fletcher and Leyffer, contributors tofilter-SQP theory have included Gould, M. Ulbrich, S. Ulbrich,Toint, Wachter, others.
The Filter approach is also used in conjunction with interior-pointmethods (see below).
Recent codes using filters:
I FilterSQP (Fletcher and Leyffer; SQP/filter)
I IPOPT (Wachter and Biegler; interior-point/filter).
Stephen Wright Continuous Optimization: Recent Developments
repeatSolve Bx(µk) for x(µk), starting from xk−1;Choose µk+1 ∈ (0, µk);Choose next starting point xk ;k ← k + 1;
until convergence.
Newton steps obtained from:
∇2xxP(x ;µ)∆x = −∇xP(x ;µ).
Primal barrier was one of the original methods for NLP (Frisch,’55; Fiacco & McCormick, ’68). Fell into disuse for various reasons,including severe ill-conditioning of the Hessian ∇2
xxP(x ;µ).(Many difficulties later turned out to be fixable.)
Stephen Wright Continuous Optimization: Recent Developments
Primal-dual approaches for linear programming were well known by’87 and were the practical approach of choice by ’90.
Extension to NLP obvious in principle, but the devil was in thedetails! Many enhancements were needed to improve practicalrobustness and efficiency.
Codes LOQO (Benson, Shanno, Vanderbei), KNITRO (Byrd et al.)and IPOPT (Wachter & Biegler) all generate steps of primal-dualtype but differ in important respects.
KNITRO and IPOPT both derive their theoretical justificationfrom the Barrier-EQP approach, but replace µ−1W 2 in thereduced system by Λ−1W . (IPOPT checks explicitly that Λ−1Wdoes not stray too far from µ−1W 2.)
Stephen Wright Continuous Optimization: Recent Developments
SoftwareIPOPT (Wachter, Biegler, ’04): Line-search filter method basedon Bw (µ), but with primal-dual steps.
Solve step equations using direct solver (MA27) forsymmetric-indefinite systems, adding perturbations to the diagonalto “correct” the inertia if necessary.
LOQO uses direct factorization of reduced step equations, withadditions to (1, 1) block to fix the inertia.
KNITRO Trust-region algorithm for Bw (µ), with “primal-dual”scaling (Λ−1W ). Merit function.
Solve step equations using an iterative method: projectedconjugate gradient.
KNITRO-Direct: Also uses direct solution of step equations anda line search on some iterations.
Stephen Wright Continuous Optimization: Recent Developments
We discuss an approach that obtains fast convergence withoutlinear independence and strict complementarity (but still needssecond-order sufficient conditions).
Denote S = (x∗, λ∗) satisfying KKT conditions.
Under our assumptions, x∗ is unique but λ∗ may not be. In fact,the set of optimal λ∗ may be unbounded.
Stephen Wright Continuous Optimization: Recent Developments
Application of MPEC: Equilibrium Problems with DesignSystems described by equilibrium conditions (e.g. economic ormechanical), with an optimization or design problem overlaid.
Example: Nash equilibrium in a 2-player game.Player 1 aims to maximize his some function f1(x1, x2) by choosing“his” variables from the set x1 ≥ 0. If f1(·, x2) is convex, theoptimal choice for a given x2 satisfies
0 ≤ x1 ⊥ −∇x1f1(x1, x2) ≥ 0.
Similarly for Player 2 (whose variables are x2), optimal choice has
0 ≤ x2 ⊥ −∇x2f2(x1, x2) ≥ 0.
The Nash equilibrium is the point (x1, x2) at which both sets ofconditions are satisfied.
Stephen Wright Continuous Optimization: Recent Developments
I Lack of CQ ⇒ Standard NLP algorithms may not work!
I Surge of interest prompted in part by ’96 monograph of Luo,Pang, Ralph.
I Various specialized algorithms proposed (’96-’01) to deal withthe special nature of the constraints.
However, Fletcher & Leyffer (’02) showed that ordinary SQP codesperformed excellently on MPEC benchmarks (and interior-pointcodes were also quite good).
Why?
Can robustness of standard NLP approaches be improved further?
Stephen Wright Continuous Optimization: Recent Developments
However for i ∈ IG ∩ IH—the biactive indices—it is less clear howthe sign of τ∗i and ν∗i should be restricted. Different flavors ofstationarity have been proposed:
I C-stationary: τ∗i and ν∗i have the same sign;
I M-stationary: Does not allow τ∗i and ν∗i of different signs, andone of them must be nonnegative.
I Strong stationarity: τ∗i and ν∗i both nonnegative.
Strong stationarity is the most appealing, as it ensures that thereis no first-order direction of descent.
Stephen Wright Continuous Optimization: Recent Developments
Optimization on the GridNew ApplicationsSoftwareModeling LanguagesInternet
Optimization on the Grid
The Grid isn’t all hype! Optimizers have done seriouscomputations on it. Some examples:
I Traveling Salesman Problem: Applegate et al. Pre-”Grid”
I metaNEOS (’97-’01): NSF-funded collaboration betweenoptimizers and grid computing groups: Condor, Globus.
I MW Toolkit for implementing master-worker algorithms ongrids
I Many interesting parallel algorithmic paradigms can beshoehorned effectively into the master-worker framework(task-farming, branch-and-bound, “native” master-worker)
I Development of MW is continuing.
Stephen Wright Continuous Optimization: Recent Developments
Optimization on the GridNew ApplicationsSoftwareModeling LanguagesInternet
Successful Implementations with MW
I Quadratic Assignment Problem: Solved the touchstoneproblem NUG30 (Anstreicher et al., ’02). Seven years of CPUtime in a week on more than 1000 computers (10 locations, 3continents).
I Two-stage stochastic linear programming (Linderoth & SW,’03). Solved problems with up to 108 constraints and 1010
variables on similar platforms. Allowed extensive testing ofstatistical properties of solutions to such problems and insightinto the nature of solutions (Linderoth, Shapiro, SW, ’02).
Stephen Wright Continuous Optimization: Recent Developments
Optimization on the GridNew ApplicationsSoftwareModeling LanguagesInternet
“New” Applications
The NLP user base is smaller than for linear programming orinteger (linear) programming, though there has been steadydemand for codes such as MINOS, CONOPT, NPSOL for manyyears.
NLP have arisen (or have attracted renewed attention) in a host ofapplications in the past 10-15 years.
Optimization with PDE constraints. An area of particular focus.Typically design or control problems overlaid on a system describedby PDEs. (see volume of Biegler et al., Springer LNCS, ’03)
Stephen Wright Continuous Optimization: Recent Developments
Optimization on the GridNew ApplicationsSoftwareModeling LanguagesInternet
Cancer Treatment Planning. Find radiotherapy treatments forcancer that deposit a given dose of radiation in the tumor whileavoiding nearby normal tissue and critical organs. Involvesoptimization problems of many types; recent interest in biologicalobjective functions give rise to NLPs.
Meteorological Data Assimilation in medium-range weatherforecasting. Special case of PDE-constrained optimization.Unknown is the state of atmosphere 48 hours ago; objectivefunction is fit between model and observations. NLP formulations(with specialized algorithms) used at all major forecasting centers.
Stephen Wright Continuous Optimization: Recent Developments
Optimization on the GridNew ApplicationsSoftwareModeling LanguagesInternet
Model Predictive Control in process engineering. Solve asequence of nonlinear optimal control problem over finite timehorizons; implement closed-loop control using open-loop strategies.Feasible SQP methods have proved to be useful.
Statistics and Machine Learning. Multicategory support vectormachines, robust estimation.
Optimization on the GridNew ApplicationsSoftwareModeling LanguagesInternet
Optimization Modeling Languages
Allows problems to be defined (and the results presented) in a waythat is natural to the application. No shoehorning into thematrices/vectors of the underlying software.
AMPL, GAMS, Maximal, AIMMS
They have become an extremely popular way to call optimizationsoftware (method of choice on NEOS Server).
Usually incorporate automatic differentiation, relieves user of needto code first derivatives by hand. (Several codes optionally allowsecond derivatives to be used; these must be provided by the user.)
They make optimzation the “outer loop”—not easy to embed theoptimization problem in a larger computation.
Stephen Wright Continuous Optimization: Recent Developments