solve multiobjective optimization: CPLEX or Matlab?

Question

I have to solve a multiobjective problem but I don't know if I should use CPLEX or Matlab. Can you explain the advantage and disadvantage of both tools. Thank you very much!

Answer 1

This is really a question about choosing the most suitable modeling approach in the presence of multiple objectives, rather than deciding between CPLEX or MATLAB.

Multi-criteria Decision making is a whole sub-field in itself. Take a look at: http://en.wikipedia.org/wiki/Multi-objective_optimization.

Once you have decided on the approach and formulated your problem (either by collapsing your multiple objectives into a weighted one, or as series of linear programs) either tool will do the job for you.

Since you are familiar with MATLAB, you can start by using it to solve a series of linear programs (a goal programming approach). This page by Mathworks has a few examples with step-by-step details: http://www.mathworks.com/discovery/multiobjective-optimization.html to get you started.

Answer 2

Probably this question is not a matter of your current concern. However my answer is rather universal, so let me post it here.

If solving a multiobjective problem means deriving a specific Pareto optimal solution, then you need to solve a single-objective problem obtained by scalarizing (aggregating) the objectives. The type of scalarization and values of its parameters (if any) depend on decision maker's preferences, e.g. how he/she/you want(s) to prioritize different objectives when they conflict with each other. Weighted sum, achievement scalarization (a.k.a. weighted Chebyshev), and lexicographic optimization are the most widespread types. They have different advantages and disadvantages, so there is no universal recommendation here.

CPLEX is preferred in the case, where (A) your scalarized problem belongs to the class solved by CPLEX (obviously), e.g. it is a [mixed integer] linear/quadratic problem, and (B) the problem is complex enough for computational time to be essential. CPLEX is specialized in the narrow class of problems, and should be much faster than Matlab in complex cases.

You do not have to limit the choice of multiobjective methods to the ones offered by Matlab/CPLEX or other solvers (which are usually narrow). It is easy to formulate a scalarized problem by yourself, and then run appropriate single-objective optimization (source: it is one of my main research fields, see e.g. implementation for the class of knapsack problems). The issue boils down to finding a suitable single-objective solver.

If you want to obtain some general information about the whole Pareto optimal set, I recommend to start with deriving the nadir and the ideal objective vectors.

If you want to derive a representation of the Pareto optimal set, besides the mentioned population based-heuristics such as GAs, there are exact methods developed for specific classes of problems. Examples: a library implemented in Julia, a recently published method.

All concepts mentioned here are described in the comprehensive book by Miettinen (1999).

Answer 3

Can cplex solve a pareto type multiobjective one? All i know is that it can solve a simple goal programming by defining the lexicographical objs, or it uses the weighted sum to change weights gradually with sensitivity information and "enumerate" the pareto front, which highly depends on the weights and looks very subjective.

You can refer here as how cplex solves the bi-objetive one, which seems not good.

For a true pareto way which includes the ranking, i only know some GA variants can do like NSGA-II.

Answer 4

A different approach would be to use a domain-specific modeling language for mathematical optimization like YALMIP (or JUMP.jl if you like to give Julia a try). There you can write your optimization problem with Matlab with some extra YALMIP functionalities and use CPLEX (or any other supported solver as a backend) without restricting to one solver.