Multi-objective optimization

Multi-objective optimization or Pareto optimization is an area of multiple-criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Multi-objective is a type of vector optimization that has been applied in many fields of science, including engineering, economics and logistics where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Minimizing cost while maximizing comfort while buying a car, and maximizing performance whilst minimizing fuel consumption and emission of pollutants of a vehicle are examples of multi-objective optimization problems involving two and three objectives, respectively. In practical problems, there can be more than three objectives.
For a multi-objective optimization problem, it is not guaranteed that a single solution simultaneously optimizes each objective. The objective functions are said to be conflicting. A solution is called nondominated, Pareto optimal, Pareto efficient or noninferior, if none of the objective functions can be improved in value without degrading some of the other objective values. Without additional subjective preference information, there may exist a number of Pareto optimal solutions, all of which are considered equally good. Researchers study multi-objective optimization problems from different viewpoints and, thus, there exist different solution philosophies and goals when setting and solving them. The goal may be to find a representative set of Pareto optimal solutions, and/or quantify the trade-offs in satisfying the different objectives, and/or finding a single solution that satisfies the subjective preferences of a human decision maker.
Bicriteria optimization denotes the special case in which there are two objective functions.
There is a direct relationship between multitask optimization and multi-objective optimization.

Introduction

A multi-objective optimization problem is an optimization problem that involves multiple objective functions. In mathematical terms, a multi-objective optimization problem can be formulated as
where the integer is the number of objectives and the set is the feasible set of decision vectors, which is typically but it depends on the -dimensional application domain. The feasible set is typically defined by some constraint functions. In addition, the vector-valued objective function is often defined as
If some objective function is to be maximized, it is equivalent to minimize its negative or its inverse. We denote the image of ; a feasible solution or feasible decision; and an objective vector or an outcome.
In multi-objective optimization, there does not typically exist a feasible solution that minimizes all objective functions simultaneously. Therefore, attention is paid to Pareto optimal solutions; that is, solutions that cannot be improved in any of the objectives without degrading at least one of the other objectives. In mathematical terms, a feasible solution is said to dominate another solution, if

, and
.

A solution is called Pareto optimal if there does not exist another solution that dominates it. The set of Pareto optimal outcomes, denoted, is often called the Pareto front, Pareto frontier, or Pareto boundary.
The Pareto front of a multi-objective optimization problem is bounded by a so-called nadir objective vector and an ideal objective vector, if these are finite. The nadir objective vector is defined as
and the ideal objective vector as
In other words, the components of the nadir and ideal objective vectors define the upper and lower bounds of the objective function of Pareto optimal solutions. In practice, the nadir objective vector can only be approximated as, typically, the whole Pareto optimal set is unknown. In addition, a utopian objective vector, such that where is a small constant, is often defined because of numerical reasons.

Examples of applications

Economics

In economics, many problems involve multiple objectives along with constraints on what combinations of those objectives are attainable. For example, consumer's demand for various goods is determined by the process of maximization of the utilities derived from those goods, subject to a constraint based on how much income is available to spend on those goods and on the prices of those goods. This constraint allows more of one good to be purchased only at the sacrifice of consuming less of another good; therefore, the various objectives are in conflict with each other. A common method for analyzing such a problem is to use a graph of indifference curves, representing preferences, and a budget constraint, representing the trade-offs that the consumer is faced with.
Another example involves the production possibilities frontier, which specifies what combinations of various types of goods can be produced by a society with certain amounts of various resources. The frontier specifies the trade-offs that the society is faced with — if the society is fully utilizing its resources, more of one good can be produced only at the expense of producing less of another good. A society must then use some process to choose among the possibilities on the frontier.
Macroeconomic policy-making is a context requiring multi-objective optimization. Typically a central bank must choose a stance for monetary policy that balances competing objectives — low inflation, low unemployment, low balance of trade deficit, etc. To do this, the central bank uses a model of the economy that quantitatively describes the various causal linkages in the economy; it simulates the model repeatedly under various possible stances of monetary policy, in order to obtain a menu of possible predicted outcomes for the various variables of interest. Then in principle it can use an aggregate objective function to rate the alternative sets of predicted outcomes, although in practice central banks use a non-quantitative, judgement-based, process for ranking the alternatives and making the policy choice.

Finance

In finance, a common problem is to choose a portfolio when there are two conflicting objectives — the desire to have the expected value of portfolio returns be as high as possible, and the desire to have risk, often measured by the standard deviation of portfolio returns, be as low as possible. This problem is often represented by a graph in which the efficient frontier shows the best combinations of risk and expected return that are available, and in which indifference curves show the investor's preferences for various risk-expected return combinations. The problem of optimizing a function of the expected value and the standard deviation of portfolio return is called a two-moment decision model.

Optimal control

In engineering and economics, many problems involve multiple objectives which are not describable as the-more-the-better or the-less-the-better; instead, there is an ideal target value for each objective, and the desire is to get as close as possible to the desired value of each objective. For example, energy systems typically have a trade-off between performance and cost or one might want to adjust a rocket's fuel usage and orientation so that it arrives both at a specified place and at a specified time; or one might want to conduct open market operations so that both the inflation rate and the unemployment rate are as close as possible to their desired values.
Often such problems are subject to linear equality constraints that prevent all objectives from being simultaneously perfectly met, especially when the number of controllable variables is less than the number of objectives and when the presence of random shocks generates uncertainty. Commonly a multi-objective quadratic objective function is used, with the cost associated with an objective rising quadratically with the distance of the objective from its ideal value. Since these problems typically involve adjusting the controlled variables at various points in time and/or evaluating the objectives at various points in time, intertemporal optimization techniques are employed.

Optimal design

Product and process design can be largely improved using modern modeling, simulation, and optimization techniques. The key question in optimal design is measuring what is good or desirable about a design. Before looking for optimal designs, it is important to identify characteristics that contribute the most to the overall value of the design. A good design typically involves multiple criteria/objectives such as capital cost/investment, operating cost, profit, quality and/or product recovery, efficiency, process safety, operation time, etc. Therefore, in practical applications, the performance of process and product design is often measured with respect to multiple objectives. These objectives are typically conflicting, i.e., achieving the optimal value for one objective requires some compromise on one or more objectives.
For example, when designing a paper mill, one can seek to decrease the amount of capital invested in a paper mill and enhance the quality of paper simultaneously. If the design of a paper mill is defined by large storage volumes and paper quality is defined by quality parameters, then the problem of optimal design of a paper mill can include objectives such as i) minimization of expected variation of those quality parameters from their nominal values, ii) minimization of the expected time of breaks and iii) minimization of the investment cost of storage volumes. Here, the maximum volume of towers is a design variable. This example of optimal design of a paper mill is a simplification of the model used in. Multi-objective design optimization has also been implemented in engineering systems in the circumstances such as control cabinet layout optimization, airfoil shape optimization using scientific workflows, design of nano-CMOS, system on chip design, design of solar-powered irrigation systems, optimization of sand mould systems, engine design, optimal sensor deployment and optimal controller design.