Depending on various types of problems there may be different answers to this question. Here are some very common types of problems.
1) linear programming problems
2) nonlinear programming problems
3) integer linear programming problems
4) integer nonlinear programming problems
5) mixed integer linear programming problems
6) mixed integer nonlinear programming problems
7) convex programming problems
8) non-convex programming problems
Kindly, name the the best suitable algorithm the optimization problems discussed here. The optimization technique may be Analytical or Metaheuristic type.
Mr. Alam
For me one of the best algorithms to solve MCDM problems is the Simplex algorithm, although it has been improved and even surpassed by genetic algorithms.
I have used a lot the Solver algorithm that is an add-in of Excel and that permits working with decimals and integer solutions as well as non-linear situations.
I am in no condition to say which is the best algorithm especially for non-linear problems, however the Simplex indicates, using the Lagrange multiplierr when it is a non-linear scenario, naturally in only the tangent point. One disadvantage of the Simplex is that it works with only one objective function and only with quantitative criteria., which in many cases is a not a very realistic assumption.
Another very important contribution of Simplex is that it simultaneously with the solution of a problem, delivers ther values of its dual, furninshing the shadow prices that have an enormous importance in sensitivity analysis
For several of these problem types there is NO one-algorithm-fits-all. For items 1 and 7 there are however interior point solvers that work extremely well. For items 3 and 5 my best advise is to try CPLEX - testing a variety of existing options regarding starting points and algorithm strategies, but to also try to reformulate your problem such that you use several variations of ways in which to define your variables, as often the more variables you have the better. Items 2 and 8 are effectively the same; there is no general tool to solve them with any guarantees, but if you have a problem with a Lipschitz continuous objective function gradient, you may be able to devise underestimators of the optimal value, for use in branch-and-bound type schemes. Items 4 and 6 are again similar, with no efficient algorithms known. The best advise is to try lots of different algorithms, with a variety of ways in which to find good starting points, such as to start with a variety of heuristic approaches, and then utilize the general solvers you have from the points where the others terminated.
Mahamad, try to tell more precisely what do you mean by "the best suitable algorithm"?
For example, some classes of integer linear programming problems may be solved by the standard simplex method (without using any "integer-specific" algorithms and, what follows, without the software which supports integer linear programming problems) but some cannot.
Nolberto, Solver is not an algorithm but a piece of software integrated with Excel (as you say later - an add-in). It contains a non-linear algorithm (the default one), the simplex method and the support of binary/integer constraints (with necessary algorithms inside). In Excel 2010 an evolutionary algorithm as well as the "alldifferent" constraint type were added. There are also some options to control the performance of the above stuff. From my experience, Solver doesn't perform well for non-linear problems.
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