         MINOS 5.5 - Description MINOS (invented by Bruce Murtagh and Michael Saunders) is a software package for solving large-scale optimization problems (linear and nonlinear programs). It is especially effective for linear programs and for problems with a nonlinear objective function and sparse linear constraints (e.g., quadratic programs). MINOS can also process large numbers of nonlinear constraints. The nonlinear functions should be smooth but need not be convex. For linear programs, MINOS uses a sparse implementation of the primal simplex method.  For nonlinear objective functions (and linear constraints), MINOS uses a reduced-gradient method with quasi-Newton approximations to the reduced Hessian. For problems with nonlinear constraints, MINOS uses a sparse SLC algorithm (a projected Lagrangian method, related to Robinson's method). It solves a sequence of sub problems in which the constraints are linearized and the objective is an augmented Lagrangian (involving all nonlinear functions). Convergence is rapid near a solution.  MINOS makes use of nonlinear function and gradient values. The solution obtained will be a local optimum (which may or may not be a global optimum). If some of the gradients are unknown, they will be estimated by finite differences. If the linear constraints have no feasible solution, MINOS terminates as soon as infeasibility is confirmed. Infeasible nonlinear constraints are difficult to diagnose. (They are treated more methodically by SNOPT.) For large problems, efficiency is improved if only some of the variables enter nonlinearly, or if the number of active constraints is nearly as large as the number of variables (i.e., if there are few degrees of freedom at a solution). MINOS can accommodate problems with many degrees of freedom (perhaps thousands), but a few hundred or less is preferable.  MINOS is suitable for general QP problems (but may find just a local optimum if the quadratic is indefinite). LSSOL and QPOPT are suitable for LP and QP problems of arbitrary size if the constraint matrix is essentially dense, but the sparse solvers MINOS and SNOPT will usually be more efficient even for dense problems. If the constraints are highly nonlinear, or the functions and gradients are expensive to evaluate, SNOPT or NPSOL may be more effective. (They use an SQP algorithm in which the sub problems have the same linearized constraints as in MINOS, but the objective is a quadratic approximation to the Lagrangian. Hence, no function or gradient values are needed during the solution of each QP. A merit function promotes convergence from arbitrary starting points.)  