1. Recent Developments in Optimization and their Impact on Control Stephen Wright Argonne National Laboratory wright@mcs.anl.gov

2. 2/9/00Aspen World 20002 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

3. 2/9/00Aspen World 20003 Nonlinear Programming (NLP) feasible region (two local solutions)

4. 2/9/00Aspen World 20004 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.

5. 2/9/00Aspen World 20005 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 www.mcs.anl.gov/otc/Guide/SoftwareGuide

6. 2/9/00Aspen World 20006 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)

7. 2/9/00Aspen World 20007 SQP Given current iteratesolve the subproblem Sequential (Successive) Quadratic Programming

8. 2/9/00Aspen World 20008 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

9. 2/9/00Aspen World 20009 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

10. 2/9/00Aspen World 200010 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.

11. 2/9/00Aspen World 200011 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

12. 2/9/00Aspen World 200012 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!

13. 2/9/00Aspen World 200013 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

14. 2/9/00Aspen World 200014 nonlinear MPC: continuous continuous nonlinear model: objective: possibly final constraint:

15. 2/9/00Aspen World 200015 nonlinear MPC: discrete discrete nonlinear model: objective: possibly final constraint:

16. 2/9/00Aspen World 200016 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)

17. 2/9/00Aspen World 200017 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

18. 2/9/00Aspen World 200018 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

19. 2/9/00Aspen World 200019 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/www.mcs.anl.gov/otc/Guide/

20. 2/9/00Aspen World 200020 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

21. 2/9/00Aspen World 200021 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

22. 2/9/00Aspen World 200022 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!

23. 2/9/00Aspen World 200023 metaneos project joint project: optimization and grid computing groups www.mcs.anl.gov/metaneoswww.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 www.cs.wisc.edu/condor Have implemented solvers for °global optimization °integer programming °stochastic optimization °quadratic assignment problem (QAP)

24. 2/9/00Aspen World 200024 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

25. 2/9/00Aspen World 200025 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

26. 2/9/00Aspen World 200026 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 QAPQAP

27. 2/9/00Aspen World 200027 MW example: QAP

28. 2/9/00Aspen World 200028 MW example: QAP

29. 2/9/00Aspen World 200029 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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34. 2/9/00Aspen World 200034 Resource profile of an MW-QAP run