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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