# Template talk:Mathematical optimization software

## Alternative taxonomies for math. opt. software

#### Mathematical optimization#Major subfields

• Convex programming [objective function: convex (minimization) or concave (maximization); constraint set: convex]
– This can be viewed either as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.
• Linear programming (LP), a type of convex programming, studies the case in which the objective function f is linear and the set of constraints is specified using only linear equalities and inequalities.
• Second order cone programming (SOCP) is a convex program, and includes certain types of quadratic programs.
• Semidefinite programming (SDP) is a subfield of convex optimization where the underlying variables are semidefinite matrices. It is generalization of linear and convex quadratic programming.
• Conic programming is a general form of convex programming. LP, SOCP and SDP can all be viewed as conic programs with the appropriate type of cone.
• Geometric programming is a technique whereby objective and inequality constraints expressed as posynomials and equality constraints as monomials can be transformed into a convex program.
• Integer programming studies linear programs in which some or all variables are constrained to take on integer values. This is not convex, and in general much more difficult than regular linear programming.
• Quadratic programming allows the objective function to have quadratic terms, while the feasible set must be specified with linear equalities and inequalities. For specific forms of the quadratic term, this is a type of convex programming.
• Fractional programming studies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem.
• Nonlinear programming studies the general case in which the objective function or the constraints or both contain nonlinear parts. This may or may not be a convex program. In general, whether the program is convex affects the difficulty of solving it.
• Stochastic programming studies the case in which some of the constraints or parameters depend on random variables.
• Robust programming is, like stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem. This is not done through the use of random variables, but instead, the problem is solved taking into account inaccuracies in the input data.
• Combinatorial optimization is concerned with problems where the set of feasible solutions is discrete or can be reduced to a discrete one.
• Stochastic optimization for use with random (noisy) function measurements or random inputs in the search process.
• Infinite-dimensional optimization studies the case when the set of feasible solutions is a subset of an infinite-dimensional space, such as a space of functions.
• Heuristics and metaheuristics make few or no assumptions about the problem being optimized. Usually, heuristics do not guarantee that any optimal solution need be found. On the other hand, heuristics are used to find approximate solutions for many complicated optimization problems.
• Constraint satisfaction studies the case in which the objective function f is constant (this is used in artificial intelligence, particularly in automated reasoning).
• Disjunctive programming is used where at least one constraint must be satisfied but not all. It is of particular use in scheduling.

In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):

Other:

• Multi-objective optimization
• Multi-modal optimization

#### Mathematical Programming Glossary: The Nature of Mathematical Programming

• Abstract
• Biconvex
• Bilinear
• Composite concave
• Continuous
• Convex
• Discrete
• Disjunctive
• Dynamic
• Factorable
• Fractional
• Geometric
• Integer
• Infinite
• Lattice
• Linear
• Mixed-Integer
• Multilevel
• Nonlinear
• Pseudo-boolean
• Reverse convex
• Semi-definite
• Semi-infinite
• Separable

#### Decision Tree for Optimization Software: Problems and Software

• Global Optimization
• Unconstrained Optimization
• Constrained Optimization
• LP/MILP
• QP/MIQP
• SDP/SOCP
• Geometric programming
• NLP/MINLP
• Network Optimization
• Constraint Programming
• Dynamic Programming / Control Problems
• Least Squares: Unconstrained LSQ, Constrained LSQ
• Complementarity
• Multiobjective Optimization
• Discrete Optimization
• ...

#### NEOS Optimization Guide: Optimization Tree and NEOS Solvers: Index

• Continuous Optimization
• Bound Constrained Optimization
• Constrained Optimization
• Derivative-Free Optimization
• Global Optimization
• Linear Programming (LP)
• Nondifferentiable Optimization
• Nonlinear Programming (NLP)
• Second Order Cone Programming (SOCP)
• Semidefinite Programming (SDP)
• Semiinfinite Programming (SIP)
• Unconstrained Optimization
• Nonlinear Least Squares (NLLS)
• Nonlinear Equations
• Discrete Optimization
• Integer Linear Programming (ILP)
• Mixed Integer Nonlinear Programming (MILNP)
• Combinatorial Optimization (CO)
• Traveling Salesman Problem (TSP)
• Optimization Under Uncertainty
• Stochastic Programming
• Stochastic Linear Programming
• Complementarity Problems and Variational Inequalities
• Complementarity Problems
• Game Theory
• Linear Complementarity Problems
• Mathematical Programs with Complementarity Constraints
• Mathematical Programs with Equilibrium Constraints
• Nonlinear Complementarity Problems
• Multiobjective Optimization
• ...

#### Optimization Software Guide: Software Classification

• Unconstrained Optimization
• Nonlinear Least Squares
• Nonlinear Equations
• Linear Programming
• Bound Constrained Optimization
• Constrained Optimization
• Network Optimization
• Integer Programming
• Misc.
• Nondifferentiable Optimization
• Multiple Criteria Optimization
• ...

#### Wiley Encyclopedia of Operations Research and Management Science: Optimization Software

• Instance Formats
• Simplex-based LP Solvers
• Interior-Point LP Solvers: BPMPD, CLP, FortMP, GIPALS32, HOPDM, GLPK, CPLEX, LINDO, LIPSOL, LOQO, MSF, MOSEK, PCx, SAS/OR, Xpress-Optimizer
• Conic Optimization [LP, SOCP, (NL)SDP] Software: CPLEX, CSDP, CVXOPT, DSDP, LMIlab, LOQO, MOSEK, PENSDP, SDPA, SDPlr, SDPT3, SeDuMi, SMCP
• MILP software: CPLEX, Gurobi, LINDO, MOSEK, Xpress-Optimizer, BLIS, CBC, GLPK, MINTO, SCIP, SYMPHONY
• MINLP software: <MIQCP> CPLEX, Xpress-Optimizer, MOSEK, SCIP; <General MINLP> alphaBB, AlphaECP, AOA, BARON, BNB, BONMIN, Couenne, DICOPT, Xpress-SLP, FILMINT, FMINCONSET, KNITRO, LAGO, LINDOBB, LINDOGlobal, LOGMIP, MIDACO, MILANO, MINLP_BB, MISQP, OQNLP, SBB
• Software for Nonlinearly Constrained Optimization: <Interior-Point Solvers> Ipfilter, IPOPT, KNITRO, LANCELOT, LINDO, LOQO; <SLP/SQP Solvers> CONOPT, FilterSQP, KNITRO, LINDO LRAMBO, NLPQLP, NPSOL, PATHNLP, SNOPT, SQPlab; <ALM Solvers> ALGENCAN, GALAHAD, LANCELOT, MINOS, PENNON

#### OPTI Toolbox: Solvers and YALMIP: Solvers

• Linear Programming (LP) / Mixed Integer Linear Programming (MILP)
• Second-order cone programming (SOCP) / Mixed Integer Second-order cone programming (MISOCP)
• Semidefinite Programming (SDP)
• System of Nonlinear Equations (SNLE)
• Nonlinear Least Squares (NLS)
• Nonlinear Programming (NLP) / Mixed Integer Nonlinear Programming (MINLP) / Global Nonlinear Programming (GNLP)

#### Optimization Online Submissions: Areas

• Combinatorial Optimization
• Complementarity and Variational Inequalities
• Convex and Nonsmooth Optimization
• Global Optimization
• Infinite Dimensional Optimization
• Integer Programming
• Linear, Cone and Semidefinite Programming
• Network Optimization
• Nonlinear Optimization
• Optimization Software and Modeling Systems
• Robust Optimization
• Stochastic Programming