In multiple-criteria decision making, multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute optimization) 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 (possibly infinite) 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 (DM).
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.
k≥2 is the number of objectives and the set n-dimensional application domain. The feasible set is typically defined by some constraint functions. In addition, the vector-valued objective function is often defined as- X; x1∈X is said to (Pareto) dominate another solution ∃i∈{1,…,k},fi(x1)<fi(x2).
A solution X∗, is often called the Pareto frontier, Pareto front, or Pareto boundary.
The Pareto front of a multi-objective optimization problem is bounded by a so-called nadir objective vector znadir=(supx∗∈X∗f1(x∗)⋮supx∗∈X∗fk(x∗))
and the ideal objective vector as
- ziutop=ziideal−ϵ,∀i∈{1,…,k} where 0"}}'>ϵ>0 is a small constant, is often defined because of numerical reasons.
Examples of applications
editEconomics
editIn 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 (more consumption of each good is preferred) 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
editIn 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 (first moment) and the standard deviation (square root of the second central moment) of portfolio return is called a two-moment decision model.
‖⋅‖ can be any L2, and 𝐲1,𝐲2∈Y it holds that u(\\mathbf{y}^2)"}}'>u(𝐲1)>u(𝐲2) if the decision maker prefers u(𝐲1)=u(𝐲2) if the decision maker is indifferent between u is obtained, it suffices to solveOptimal control
editThe lexicographic method assumes that the objectives can be ranked in the order of importance. We assume that the objective functions are in the order of importance so that ming(f1(x),…,fk(x),θ)s.t.x∈Xθ
where θ, and wi>0 are the parameters of the scalarization.
- ϵj are parameters as above and ρ∑i=1kfi(x)zinadir−ziutop is called the augmentation term, 0"}}'>ρ>0 is a small constant, and z¯ representing objective function values preferred by the decision maker.
- Sen's multi-objective programming
- r and minimization minx∈Xmaxifi(x)wi
- where the weights of the objectives 0"}}'>wi>0 are the parameters of the scalarization. If the parameters/weights are drawn uniformly in the positive orthant, it is shown that this scalarization provably converges to the Pareto front, even when the front is non-convex.
Smooth Chebyshev (Tchebycheff) scalarization
editThe smooth Chebyshev scalarization; also called smooth Tchebycheff scalarisation (STCH); replaces the non-differentiable max-operator of the classical Chebyshev scalarization with a smooth logarithmic soft-max, making standard gradient-based optimization applicable. Unlike typical scalarization methods, it guarantees exploration of the entire Pareto front, convex or concave.
- Definition
For a minimization problem with objective functions guSTCH(x∣𝝀)=uln(∑i=1kexp(λi[fi(x)−ziideal]u)),u>0,𝝀∈Δk−1,
where Δk−1.
As u controls the trade-off between differentiability and approximation accuracy: smaller values yield a closer match to the classical Chebyshev scalarisation but reduce the Lipschitz constant of the gradient, while larger values give a smoother surface at the cost of looser approximation.
STCH covers whole Pareto front; convex or concave; because for every preference vector guSTCH is continuously differentiable with an ε-optimal point is reachable in 𝒪(1/ε2) iterations. - Pareto optimality — For any 0"}}'>u>0 every minimizer of u∗>0 such that, for 𝝀; when the Pareto front is convex this holds for all 0"}}'>u>0.
For example, portfolio optimization is often conducted in terms of mean-variance analysis. In this context, the efficient set is a subset of the portfolios parametrized by the portfolio mean return μP; see Mutual fund separation theorem for details. Alternatively, the efficient set can be specified by choosing the portfolio shares to maximize the function
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