Recent Developments in Optimization

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Recent Developments in Optimization

• and their Impact on Control
• Stephen Wright
• Argonne National Laboratory
• wright_at_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)
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/G
  uide/SoftwareGuide

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NLPSQP

• 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
Sequential (Successive) Quadratic Programming
Given current iterate
solve the subproblem
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NLP Augmented Lagrangian

• Known since mid-1970s, but serious implementation
  not attempted until 1988.
• fairly robust, good global convergence properties
• its easy to implement a simple version
• efficiency varies
• code Lancelot

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NLPLog 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

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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)
• Its hard to interoperate with other numerical
  code, particularly linear algebra code.
• Its 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

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nonlinear MPC continuous
continuous nonlinear model
objective
possibly final constraint
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nonlinear MPC discrete
discrete nonlinear model
objective
possibly final constraint
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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)

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

• Case studies, including
• diet problem
• portfolio optimization
• simplex applet
• Software Guide
• Optimization Tree outline of various algorithms
• FAQs for linear and nonlinear programming

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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