I think it depends on the relationship of your parameters. Or maybe you can maximize the whole utility which consists of all parameters.

Thank you Fang. I am interested in doing a coupled optimization. I want to minimise an objective function while maximising another function all tethered to constraints. Any ideas?

Not sure i get your question correctly, but maybe you can take a look at the Hybrid Game and HAPMOEA papers below:

http://www.deepdyve.com/lp/elsevier/hybrid-game-strategies-for-multi-objective-design-optimization-in-WW7GfwFK0I

https://www.researchgate.net/publication/228575944_Multidisciplinary_Design_Optimisation_Of_Joint_UCAV_Using_HAPMOEA_with_Uncertainty

Article Multidisciplinary Design Optimisation Of Joint UCAV Using HA...

I thin both the minimize and maximize contribute a common utility to your system. All parameters have the same final outcome to your model. You can transform them into the same outcome. For example, you want to minimize the cost, and it equals to maximizing the profit. All other parameters can be evaluated or calculated related to the profit. So, final objective is to maximize the profit.

Dear Cristopher,

In finance we use Mean-Variance model to maximize the return and minimize the risk. Maybe this model will give you some idea for your further work. But the main disadvantage of this model is that is static model.

I think that better approach is to use stochastic programming approach and as Fang suggested to maximize the profit. Because, if your are analysing market which is not characterised as a monopoly or oligopoly than the cost is equal to the marginal cost. This is the minimum.

Please check out multi-criteria optimization. Obviously you cannot maximize/minimize two things simultaneously, as the two may be mutually inconsistent with each other. But it may be possible to strike the right balance between the two based on some suitable criterion. Topics like goal programming seek to do just that.

I think multi-criteria optimization is an option but I'm not sure what you meant by regularization parameters.

One way is to put both functions in your objective function, with one of them -ve. i.e. if you want to keep it as a minimization problem, you could add the maximization function with a negative coefficient. You can also add weights for each sub-objective after some form of normalizing each function of your objective.

Your problem is multi-objective, and you are probably looking for the whole set of pareto-optimal solutions. You have not stated anything about your model. If you have a MIP model, you can apply two-phase optimization. This requires some programming and a god MIP solver (CPLEX or Gurobi). Alternatively you can try to find the pareto-front heuristically, typically using an evolutionary algorithm. I can recommend the PISA framework:

http://www.tik.ee.ethz.ch/pisa/

Thank you for all the answers. I am using CPLEX and LP at the moment, but MIP is possible. I believe it is multi-objective too. I cannot go into too much detail about the model unfortunately.

I will test a few things out and maybe state some on here to see what you all think!

Chris, you can actually do this by having a if...then formula link to a cell that feeds into the first objective function...then have the second one as a constraint with a Max{} or Min{} specification ... and then invoking the Solver via VBA or simply Excel...hope this helps ...essentially it is an application of the saddle point Kuhn-Tucker min-max concavity and optimization theorem. Cheers!

As Yogesh stated mathematically it's not possible, see the paper below.

Article Theory of games and economic behavior (60th anniversary comm...

You can maximize and minimize multi-objective function, it is okey.

But not optimizing variables. The design variables are just tools which are being manipulated to get the optimum solution.

maximization is opposite to minimization, just add a negative sign.

I think there are some serious terminology issues her: (a) what is a "large scale optimization procedure"? (b) what does "optimize two parameters" actually mean (typically the goal is to minimize/maximize the value of a scalar valued criterion function whose value depends on adjustable parameters) (c) what are regularization parameters?

i am also agree@Thomas Stidsen please find attachment

Yes, we can do multi-objective optimization. We can use heuristics to solve such problems.

Could you clarify the conditions that characterize the "solution' that you locate to a multi-objective optimization problem?

The optimum solution for one objective may not coincide with the second objective. You have to consider the pareto-optimal solutions, i.e. the set of solutions that are on the efficiency frontier. These are the set of solutions that are not dominated by any other feasible solution, i.e. there is no solution in which both objectives are better. If you have an upper or lower bound for one objective, then you can add that as a constraint in your formulation to obtain the optimal solution with respective to the other objective that satisfies the upper or lower bound. Otherwise, you are looking at a set of solutions that are not any better than one another, and perhaps pick the one that is the most preferable according to some other criteria.