# Suggestion for development of optimization solvers under MPT

Pavel,

Given as you say (in https://mct.userecho.com/communities/1/topics/125-mp-linprog-support) the high demand for optimization solvers and the complexity and difficulty of implementing high quality optimization solvers "from scratch", i.e., without access to existing double precision code as starting point for modification, I suggest a different development approach.:

Work with developers//vendors of high-end optimization solvers to modify their code to produce an MPT version. Perhaps the vendor would agree to make such a version available to its customers at no extra cost. What would be in it for the vendor would be potential extra sales. This could still be a non-trivial effort to do well, because even though solvers have a large number of settable parameters related to various tolerances, there may be some algorithmic and implementation decisions which might be implicitly predicated on an assumption that the code is double precision. I have no idea of the do-ability of an MPT version, for instance, if the solver code is in C or C++ and the vendors produce mex file, as I think is the case for the following.

Some suggestions - these are all top-end, under active development, and all have MATLAB toolbox versions.

Note: LP = Linear Program, QP = (linearly constrained) Quadratic Program. QCQP = Quadrqatically Constrained Quadratic Program.. Mi = mixed integer. SOCP = Second Order Cone Problem, which subsumes LP, convex QP and convex QCQP. LMI = Linear Matrix Inequality. = Linear Semidefinite Program.

CPLEX (see https://www.ibm.com/support/knowledgecenter/en/SSSA5P_12.6.3/ilog.odms.cplex.help/CPLEX/MATLAB/topics/gs_ov_toolbox.html ) - LP, MILP,, convex QP, MIQP, convex QCQP, convex MIQCQP, SOCP, MISOCP, local optimal and globally optimal solution of non-convex QP and MIQP, drop-in replacement for lsqlin (linearly constrained linear least squares)

Mosek - LP, MILP, convex QP, convex MIQP, SOCP, MISOCP, LMI, exponential cone (in forthcoming Mosek version 9.0)

KNITRO -all problem classes handled by FMINCON + integer constraints + special handling of complementarity constraints (MPEC) and SOC constraints. Higher performance and more robust than corresponding FMINCON algorithms.

Gurobi - same problem classes as CPLEX, except no non-concex QP and MIQP capability

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I am afraid, that this suggestion is a bit out of scope for MCT. All above mentioned commercial solvers are realized typically at C/C++ and available Matlab toolboxes (all supports only double-precision class) plays only a role of interface between these low-level solvers (implemented as C/C++ MEX files) and Matlab environment.

The only way, in this case, is incorporation of multi-precision computing libraries (like MPIR C Library for Multiple Precision Integers and Rationals, MPFR C Library for Multiple-Precision Floating-point computations with correct Rounding, MPC C Library for the Arithmetic of Complex numbers with arbitrary high precision, MPFR C++ multi-precision floating point number class, etc.), to these low-level solvers source code, which is definitely not simple task at all.

By my best knowledge of MCT, the only possible way, how to perform multi-precision optimization via MCT is modification of suitable optimization solvers, which are completely implemented at MATLAB to use mp class. But, I am sure (!!!), that for above mentioned commercial solvers are not available any strictly MATLAB coded solvers.

Dear Mark,

Thank you very much for your suggestion and for summary on optimization solvers. I will use it as a reference.

In order to support multiple precision, core C/C++ code of each solver must be modified.

The minimal modification would be just changing the data types, constants, tolerances, etc. However in most of the cases these codes also rely on third-party libraries like Intel MKL/LAPACK, SuiteSparse or even some outdated FORTAN libraries (like NAG). This part must be modified to call analogous (if present) functions from MCT (also on C/C++ level).

Overall this work requires tight collaboration with those vendors with access to C/C++ codes on both sides.

We are open for such collaboration, but from my experience, other companies are not ready to invest because market is really tiny and it won't pay off.

For example, I talked to Maple top managers few years ago. They said that fast multiple precision is low priority task because of low demand.

P.S.

To my knowledge, developers of some of the libraries you mentioned (Mosek, CVX) are using MCT in their development work.

Well, I didn't think it would be easy, but perhaps much easier than writing a high quality optimization solver from scratch. There are a tremendous amount of non-obvious subtleties which make a huge difference in actual performance and reliability, and that is not easy to do.

SDPA is available in the variant SDPA-GMP (SDPA with arbitrary precision arithmetic);) http://sdpa.sourceforge.net/index.html . However, SDPA is not the most reliable LMI solver (not in Mosek's league).

quadMINOS is a quadrupke (not arbitrary) precision version of MINOS which is available commercially https://www.gams.com/latest/docs/S_MINOS.html . See also https://web.stanford.edu/group/SOL/talks/14beijingQuadMINOS.pdf , including FAQ on the last page, which as of Dec 2016, states there is no MATLAB version of quadMINOS, and also that there is quadSNOPT (in f90 snopt9 we can change 1 line), i..e, quad precision version of the SNOPT continuous variable smooth nonlinear optimization solver.

As for optimization solvers written in MATLAB, there is the free solver PENLAB, which can handle nonlinear SDPs,(none of the solvers listed in my post above address nonlinear SDPs, to include Bilinear Matrix Inequalities (BMIs), and nonlinear constraints such as FMINCON addresses. It is a free, lower performing version of the commercial solver PENNON, now distributed via NAG in the NAG C library.. Unfortunately, PENLAB is riddled with bugs, and frequently hangs, unlike PENNON, or its other sibling, PENBMI (addresses BMIs rather than fully general nonlinear SDPs), bath written in C/C++, and having many fewer bugs than PENLAB. Supposedly a new version of PENLAB is under development, or is at least being considered for such, but I don't know its statues,, I wouldn't invest much effort in trying to make an arbitrary precision version of PENLAB until there's a relatively bug-free, or at least "usable" version available as a starting point. A version of PENNON packaged to be called from MATLAB will supposedly be available in NAG Toolbox Mark 27 in summer 2019 - however as of 2 years ago, it was going to come out 1.3.4 years ago, so who knows. Although PENLAB and PENNON address standard nonlinear continuous nonlinear programming (in addition to SDP constraints, they can't hold a candle to KNITRO when solving problems addressable by KNITRO (or FMINCON)., so they are not a good choice except for problems having SDP constraints, possibly in addition to other constraints,

Also I forgot to mention in my post above that KNITRO also has an SR1 Quasi-Newton option, in addition to BFGS and BFGS-L (LBFGS), whereas FMINCON does not have offer an SR1 option. The SR1 update, when used in a rust region, as in KNITRO, has better capability to handle sustained non-convexiities in the Lagrangian on the path to the optimum,on account of not enforcing (maintaining) positive semoidefiniteness of the approximation of the Hessian of the Lagrnagian as does BFGS, SR1 can quickly adjust its Hessian eigenvalues to match the curvature it encounters. But if your problem is convex, you're probably better off with BFGS.

Thank you for sharing your deep knowledge - very helpful.

I quickly checked the source code of SDPA-GMP.

Indeed, there are many weak points. It uses MPACK which is very buggy and unstable conversion of some LAPACK functions to arbitrary precision. I tested it myself few years ago - that was nightmare. E.g. half of the indices in the SVD code was hand-translated from FORTRAN to 0-based indices, another half still uses FORTRAN 1-based indices..... Not sure how people use it at all. I ended up writing my own MP-LAPACK with parallelism - it is part of toolbox now.

Another issue is that SDPA-GMP relies on Spooles for sparse matrices factorization. This is way too suboptimal, something like UMFPACK/CHOLMOD would be much better alternative. Also I would use something like iterative refinement to make sure our matrix solution has relative accuracy comparable to eps.

As for algorithm itself, I am not sure why code relies on Cholesky decomposition which is unstable in a presence of ill-conditioning. It already breaks when cond(A) ~ 1/sqrt(eps). Rank revealing (or at least with decent column pivoting) QR would be much more stable, it can handle ill-conditioning up to cond(A) ~ 1/eps. If our goal is accuracy, then algorithm must seek the most reliable way to handle ill-conditioning. But I don't see this in SDPA-GMP.

QUADMINOS looks interesting but I am not sure if I can access the source code.

Thanks for sharing all the links - I will study them (wasn't aware of SR1 option in KNITRO).

What solvers would you suggest to have in arbitrary precision (by priority)?

In some sense of decreasing priority (but other people have different priorities I'm sure):

I'd pick KNITRO. I suppose arbitrary precision need not be implemented for all algorithm options of which there are several https://www.artelys.com/tools/knitro_doc/2_userGuide/algorithms.html ,. combined with derivative options https://www.artelys.com/tools/knitro_doc/2_userGuide/derivatives.html .

Also, Mosek which can address LMIs and other conic problems (in forthcoming Mosek version 9, any product of Second Order Cones, semidefinite cones, exponential cones, and power cones, with exponential cones and power cones being new for Mosek 9). As you know with your eigenvalue work, sometimes you have an inherently ill-conditioned problem, and need to solve semidefinjite problems to high precision. I really don't like it when solvers cheat on the semidefinite constraints allowing slightly negative eigenvalue for psd constraint, and I don't like having to build in buffer against that by constraining to be >= small_number_as buffer_against_cheating). There actually is VSDP,http://www.ti3.tu-harburg.de/jansson/vsdp/VSDP2012Guide.pdf which is built on top of INTLAB, but I haven';t heard of anyone using VSDPt to practical effect.

CPLEX ,might be a good choice, but IBM is bureaucratic as hell, so good luck with that. Gurobi (smaller problem class than CPLEX, since it will not do non-convex QP or non-convex MIQP), also has top reputation for what it does, and was started by the founder of CPLEX after CPLEX was sold.

Arbitrary precision seems like it might not be useful for tough mixed integer problems (except when continuous relaxations are extremely ill-conditioned) because you are usually limited by long run-time to specify a non-trivial gap, such as 1% or higher,(as termination criterion)( between upper bound from best integer feasible solution and lower bound from continuous relaxation. But with arbitrary precision, you can reduce the integrality tolerance, allowing Big M logic constraints to have larger M (bound on variable) and have logic still function correctly. Many naive users specify a very large Big M value, which given default integrality tolerance means that their Big M constraints might not function as intended, thereby producing the wrong answer. But for certain very tough problems, the M needs to be very big, which means that the integrality tolerance needs to be very small, which can be achieved with arbitrary precision,.

Of course, there will be a huge performance penalty from doing floating point in SW. I think some folks have looked at hybrid implementations which implement higher precision only in vital places, and use HW floating point for everything else.

But cooperation with any commercial optimization solver developers is always very problematic task. My personal experience in this domain is very negative. Any idea how will the final product look like?