Recent Developments in Optimization

and their Impact on Control

Stephen WrightArgonne National Laboratory

wright@mcs.anl.gov

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outline

• algorithms for nonlinear optimization• software tools for structured nonlinear

optimization• applications to nonlinear control• web resources: NEOS Guide and Server• solving problems on workstation networks• computational steering and monitoring through

browser interfaces

www.mcs.anl.gov/~wright/papers/aw2000.ppt

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Nonlinear Programming (NLP)

.for,0)(,for ,0)(subject to )( min

EixcIixcxf

ii

x

feasible region

(two local solutions)

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How is NLP used?

• many real problems are really nonlinear; linear or quadratic approximations cannot give useful results.

• applications in process engineering include:° set point calculation,° nonlinear model predictive control,° process design.

• when integer variables are present, NLP algorithms are embedded in a higher-level algorithm, e.g. branch-and bound.

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NLP algorithms and codes

• less advanced than LP codes• difficult to design a completely robust code,

because NLP paradigm is so broad• global minimizer is not guaranteed in general!• there is a wide range of general purpose codes and

algorithms• can be adapted to structure of specific applications

(some algorithms/codes more easily than others)• See NEOS Guide for pointers: www.mcs.anl.gov/

otc/Guide/SoftwareGuide

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NLP:SQP• Many variants

° “reduced space” variant useful for process control applications, where state transition equations, mass balance constraints, etc, make the true dimension of the parameter space small.

• Excellent local convergence properties° But needs enhancement to achieve convergence

to a stationary point° Some forms need second derivatives° Code: SNOPT (new, good for large-scale

problems)

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SQP

Given current iterate solve the subproblemkx

IiT

kiki

EiT

kiki

kTT

kp

pxcxc

pxcxc

pBppxf

,0)()(

,0)()(21)(min

Sequential (Successive) Quadratic Programming

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NLP: Augmented Lagrangian

• Known since mid-1970s, but serious implementation not attempted until 1988.° fairly robust, good global convergence

properties° it’s easy to implement a simple version° efficiency varies° code: Lancelot

)(1

)(21)()();(min

1

2

xcvv

xcxcvxfxL

kk

Tkkx

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NLP:Log Barrier

• known since mid-1960s, but never adopted in production code because of problems with ill-conditioning of subproblems

• recent renewed theoretical study, due to connection with interior-point methods

• some aspects are used in primal-dual interior-point algorithms

0

)(ln)();(min

xcxfxPIi

ix

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NLP: Primal-Dual Interior-Point

• extensions of the most successful interior-point methods for LP to NLP

• many conceptual difficulties due to nonconvexity, need for guaranteed convergence to a stationary point

• intense research (theory and practice) for past 5 years, but no obvious winning approach yet

• PDIP adapt well to structured problems (e.g. model predictive control), as in quadratic case

• long-term prospects still good, but much more experimentation and design work is required.

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designing customizable NLP software

• Current optimization codes mostly follow traditional mathematical software design° usually written in FORTRAN° subroutine-call interface (though interfaces in

AMPL, GAMS now also available)• It’s hard to interoperate with other numerical code,

particularly linear algebra code.• It’s difficult to

° exploit problem structure to improve efficiency ° exploit developments in linear algebra codes° implement on advanced (parallel) computers

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OOQP: object-oriented QP code• object-oriented solver for convex quadratic

programming° uses interior-point algorithm° motivation: many apps (including linear model

predictive control!) each with special structure° can specialize the linear algebra methods, or use

general sparse methods, while re-using the same top-level code

° interoperates with cutting-edge linear algebra software (LAPACK, PETSc, SuperLU)

° can be embedded in NLP codes, which often need to solve QP subproblems!

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nonlinear MPC

• given a nonlinear process model, decide on the control at a given time by solving an open-loop problem over the finite interval

• to ensure nominal stability, may impose a state constraint at the endpoint, e.g.

• in principle, MPC is good at handling state and control constraints, e.g.

• good theory and algorithms are available for the case of a linear model, but the nonlinear case is more complex

jt],[ Ttt jj

0)( Ttx j

UuXx ,

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nonlinear MPC: continuous

],0[,)(,)(

,)0(),,(

TtUtuXtx

xxuxFx j

continuous nonlinear model:

T

dttutxL0

))(),((minobjective:

possibly final constraint:.)(or0)( WTxTx

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nonlinear MPC: discrete

NkUuXxxxuxfx

kk

kkkk

,,1,0,,,),,( 01

discrete nonlinear model:

)(ˆ)(min 10

,

Nk

N

kk xLuxL

objective:

possibly final constraint:.or0 11 Wxx NN

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applying NLP techniques• the discrete MPC formulation can be viewed as

an NLP in which° variables are° plant (state equation) is an equality constraint° additional constraints from endpoint condition

and• nice structure: derivative matrices block-diagonal • most algorithms can exploit this structure in

principle-but in practice?• in SQP, the QP approximation is just a linear MPC

problem (but may not be convex)

kk ux ,

UuXx kk ,

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stability and practical strategies

• under mild conditions on L(x,u), feasibility of the nonlinear MPC subproblem implies stability in the nominal case (Scokaert, Mayne, Rawlings)

• in nominal case, the solution from previous timepoint, after shifting, is still optimal

• in practice, upsets and model inexactness make it necessary to reoptimize, but the previous solution can be used as a “hot start” point

• “suboptimal” strategies can be applied that do not require the reoptimization to be exact

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role of NLP solvers in MPC

• NLP solvers should be able to° exploit structure (for efficiency)° take advantage of a hot start° be embedded in some “global optimization”

framework, that periodically checks to see if there is a better strategy that is somewhat removed from the current part of the solution space.

• Also can use NLP solvers to estimate the set W in a dual-mode strategy

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NEOS Guide

• Case studies, including° diet problem° portfolio optimization° simplex applet

• Software Guide• Optimization Tree: outline of various algorithms• FAQs for linear and nonlinear programming

Web site with information for optimization users of all kinds: students, the curious, and motivated,experienced users. www.mcs.anl.gov/otc/Guide/

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remote problem-solving• A new way to interact with numerical software is

to construct models / define problems locally on a PC / workstation, but have the solution computed remotely on a compute server

• Use the Internet as a compute engine for problem solving, not just as an informational resource

• nice interfaces are important• users avoid need to purchase hardware and

software, upgrade• good for benchmarking too

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NEOS Server• Extensive solver coverage, public and proprietary

° linear programming ° integer programming ° nonlinear programming ° complementarity, stochastic programming

• Submit jobs through email, web, or unix socket• Users supplies model and data in standard

formats, or AMPL• Server schedules job on machines in various

places (Argonne, NU, Wisconsin, Arizona)• www-neos.mcs.anl.gov

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metacomputing platforms• use networks of PCs or workstations as a

computational platform• much cheaper than a parallel computer; uses

resources that are otherwise wasted• need for a software infrastructure to make this

messy environment a usable one:° scheduling, allocation tools° parallel programming tools (MPI, PVM)° persistency tools to ensure job completion

• algorithms and applications must match the platform - but a surprising number of them do!

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metaneos project• joint project: optimization and grid computing groups

www.mcs.anl.gov/metaneos° build software infrastructure for optimization° implement optimization algorithms ° solve big problems cheaply!

• Use Condor to manage the compute resources: www.cs.wisc.edu/condor

• Have implemented solvers for° global optimization° integer programming° stochastic optimization° quadratic assignment problem (QAP)

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MW: master-worker API for Condor• Many parallel algorithms follow the master-worker

paradigm:° Master maintains a pool of work units (tasks),

sends tasks to workers and processes the results of completed tasks

° Workers receive tasks from master, computes results and sends them to master, waits for another task

• Branch-and-bound algorithms for mixed-integer LP and NLP can be “mapped” to this paradigm. Also for specailized combinatorial problems such as quadratic assigment

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MW• Master runs outside Condor pool; Workers execute on

processors inside the pool (currently up to about 200, but varies continually)

• MW provides a framework for specifying° how the Master manages the task pool° what information constitutes a “task”, i.e. is sent to

the workers° what initializing information a new worker i sent

when it becomes available° how the Master processes results from a completed

task° what statistics to maintain

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MW example: QAP

• Given a set of° n locations° n factories° flows of given amount between some factory pairs

• Decide how to assign factories to locations in a way that minimizes the total of (flow x distance)

• There are n! possible combinations (grows faster than exponential!) so need smart heuristics to home in on the interesting possibilities

• see NEOS Guide Case Studies for details on QAP

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MW example: QAP

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MW example: QAP

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iMW• Web-based problem-solving environment for

solvers running on grid computing platforms.• Supports remote submission, monitoring,

steering of jobs.• Uses Corba to communicate with master object

° monitor execution° steer (vary algorithm parameters during

execution)° suspend applications.

• Uses XML to produce “dynamic” web pages with status information.

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Resource profile of an MW-QAP run