vecmin f(x) subject to h(x)=0, g(x)>=0,
n=dim(x), m=dim(g), p=dim(h).
Many problems in real life are multiple criteria decision problems. These are usually solved by proper scalarization and parametrization. The vecmin means usually a Pareto optimum. An optimum in the sense of the natural halforder of R^n need not exist.
Solvers which explicitly provide for multiple objectives are
|SQP method for differentiable constrained NLPs, AMPL interface (Matlab)
|solver for multi-criteria integer programming and multi-criteria linear programming with an arbitrary number of objectives
|evolutionary method (Matlab)
|reactive search, free for research
|Parallel Global Multiobjective Optimizer (C++/Python)
|Overview of commercial and other MCDM software
|Multiobjective optimization framework in Java
|An SQP variable metric method for minimax optimization, linear constraints.
|uses ODEs to generate search direction followed by Newton to correct back to feasible region, authors claim it is very robust but somewhat slow.
|to be used interactively through a web-interface ,allows nonsmooth problems, uses bundle method
|Normal Boundary Intersection (Matlab)
|extensive Multiple-Objective MetaHeuristics library in Java
|Evolutionary Multi-objective Optimization (Python, MPI)
|Strength Pareto Evolutionary Algorithm (C++)
|SPEA implementation for the TSP (C++)
|Decision analytic tool for Windows 95/98/NT
|Linear and nonlinear problems; feasible/resonable goals method. Windows binaries.