18th Mar, 2023

University of Sharjah

Discussion

Started 5th Sep, 2022

Why use metaheuristic algorithms when there are so many mathematical optimization tools available, like GAMS?

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Ankur Maheshwari Very open question, so likely impossible to answer completely, but from a practitioner point of view, you'd use meta (and hyper) heuristics when your search space is either too vast, or irregular, and you can only specify partial/incomplete traits of an optimal solution(s), but not a full model specification (e.g. 'I'll know it when I see it').

To put it differently, metaheuristics work with very limited information about the solution/search space, whereas (domain specific) optimization can exploit more information, with subsequent tighter convergence.

As a hypothetical example, say you're solving a hard engineering optimization problem with differential evolution (DE), only to realize you can actually frame it as an integer programming problem. In that case, specific IP solvers will likely be vastly superior in convergence, because they exploit what DE cannot.

But as I said earlier, "mathematical optimization tools" is very generic, so depending on your interpretation there can be significant overlap.

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Ankur Maheshwari I agree with Ben Cardoen that a heuristic/partial search algorithm may provide a sufficiently good solution to an optimization problem, especially with incomplete or imperfect information or limited computation capacity where commercial solvers may struggle. But the key issue of heuristics is that they do not guarantee that a globally optimal solution can be found on some class of problems when compared with commercial solvers such as Ipopt, Cplex, Mosek, Gurobi,..., for NLP or MILP or MINLP problems.

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Dear Ankur Maheshwari,

According to Dr. Ben Cardoen and Dr. Ghulam Mohy-ud-din answering this question is hard, but it's easy to understand if you can understand the **objective function**.

The objective function is a means to maximize (or minimize) something. This something is a numeric value. In the real world, it could be the cost of a project, a production quantity, profit value, or even materials saved from a streamlined process etc.

There are lots of optimization problems in fields of researches which have varying Objective Functions. The main mathematical optimization tools (As you mentioned) are Gradient-based, which requires the objective function to be Derived and some of them are based on stochastic search (searching around the search space randomly), but as you know:

-Some objective functions cannot be Mathematically modeled or they are too hard to be modeled Mathematically so researchers are unable to use Gradient-Based methods.

-Some objective functions can be modeled mathematically but they are overly complex, like NP-Hard problems and etc. So using mathematical methods is not logical.

-Some objective functions are Non-Monotonic, so the stochastic methods might not find the right answer for the optimization problem (because they cannot escape local optima).

-Some objective functions are Discontinuous, Non-Differentiable etc. so again Gradient-Based methods are unusable.

So here is where the Heuristic or Meta-Heuristic Optimization Algorithms help researchers.

These algorithms are designed to find the global optima and escape from local Optimas. These designs are named by "Exploration" and "Exploitation" in which Exploration refers to searching the unexplored area of the feasible region while exploitation refers to the search of the neighborhood of a promising region. One of the challenges of optimization algorithms is to find the appropriate balance between Exploration and Exploitation.

Sincerely yours,

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