Multiobjective optimization

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Multi-objective optimization (or programming)[1] [2] also known as multi-criteria optimization, multi-attribute optimization, is the process of simultaneously optimizing two or more conflicting objectives, subject to certain constraints. Multiobjective optimization problems can be found in various fields: product and process design, finance, aircraft design, oil and gas industry, automobile design, or wherever optimal decisions need to be taken in the presence of trade-off between two conflicting objectives. For example, maximizing profit and minimizing cost of a product; maximizing performance and mimimzing fuel consumption of a vehicle; minimizing weight and maximizing the strength of a particular component, etc. If a multiobjective problem is well formed, there should not be a single solution that simultaneously minimizes each objective to its fullest. In each of these examples, we are looking for a solution for which we know that each objective has been optimized to the extent that if we try to optimize it any further, then the other objective(s) will suffer as a result. Finding such a solution, and quantifying how much better this solution is compared to other such solutions (there will generally be many) is the goal of setting up and solving a multiobjective optimization problem.


Contents

[edit] Introduction

In mathematical terms, the multiobjective problem statement can be written as:

\begin{align} \min_{x} &\left[\mu_1(x), \mu_2(x), ..., \mu_n(x) \right]^T & \\ s.t. & \\ g(x) & \le 0 \\ h(x) & = 0 \\ x_l \le & x \le x_u \end{align}

where μi is the i-th objective function, g and h are the inequality and equality constraints, respectively, and x is the vector of optimization or decision variables. The solution to the above problem is a set of Pareto points. Pareto solutions are those for which improvement in one objective can only occur with the worsening of at least one other objective. Thus, instead of a unique solution to the problem (which is typically the case in traditional mathematical programming), the solution to the above problem is an infinte set of Pareto points.

Pareto optimal solution: A design point in objective space μ * is termed Pareto optimal if there does not exist another feasible design objective vector μ such that \mu_i \leq \mu_i^* for all i \in \left\{ {1,2,...,n } \right\}, and \mu _j < \mu_j^* for at least one index of j, j \in \left\{ {1,2,...,n } \right\}.

[edit] Solution Methods

  • Constructing a single aggregate objective function (AOF)

This is perhaps the most intuitive approach to solving the multiobjective problem. The basic idea is to combine all of the the objective functions into a single functional form, called the AOF. A well-known combination is the weighted linear sum of the objectives. One specifies scalar weights for each objective to be optimized, and then combine them into a single function that can be solved by any single-objective optimizer (such as SQP, pattern search etc.). Clearly, the solution obtained will depend on the value (more correctly, the relative value) of the weights specified. For example, if we are trying to maximize the strength of a machine component and minimize the production cost, and if we specify a higher weight for the cost objective, compared to the strength, our solution will be one with a lower cost than higher strength. The solutions obtained using the weighted sum are always Pareto optimal, but coming up with meaningful combinations of weights can be challenging.[3]

  • Normal Boundary Intersection (NBI) method [4]
  • Normal Constraint (NC) method
  • Multiobjective Genetic Algorithms (MOGA) [5]

[edit] See also

[edit] References

  1. ^ R. E. Steuer. Multiple Criteria Optimization, Theory Computations and Applications. John Wiley & Sons, Inc., New York, 1986.
  2. ^ Y. Sawaragi, H. Nakayama, and T. Tanino. Theory of Multiobjective Optimization, vol. 176 of Mathematics in Science and Engineering, Academic Press Inc., Orlando, FL, 1985.
  3. ^ A. Messac, A. Ismail-Yahaya, and C. A. Mattson. The Normalized Normal Constraint Method for Generating the Pareto Frontier. Structural and Multidisciplinary Optimization, 25(2):86–98, 2003.
  4. ^ I. Das and J. E. Dennis. Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems. SIAM Journal on Optimization, 8:631–657, 1998.
  5. ^ K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan. A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2):182–197, 2002.
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