Optimization (mathematics)

"Cost function" redirects here. For the economics concept of a cost function, see cost function (economics).

The maximum of a paraboloid

In mathematics and computer science, optimization, or mathematical programming, refers to choosing the best element from some set of available alternatives.

In the simplest case, this means solving problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. This formulation, using a scalar, real-valued objective function, is probably the simplest example; the generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. More generally, it means finding "best available" values of some objective function given a defined domain, including a variety of different types of objective functions and different types of domains.

Contents

History

The first optimization technique, which is known as steepest descent, goes back to Gauss. Historically, the first term to be introduced was linear programming, which was invented by George Dantzig in the 1940s. The term programming in this context does not refer to computer programming (although computers are nowadays used extensively to solve mathematical problems). Instead, the term comes from the use of program by the United States military to refer to proposed training and logistics schedules, which were the problems that Dantzig was studying at the time. (Additionally, later on, the use of the term "programming" was apparently important for receiving government funding, as it was associated with high-technology research areas that were considered important.)

Other important mathematicians in the optimization field include:

  • Hoang Tuy
  • Richard Bellman
  • Ronald A. Howard
  • Leonid Kantorovich
  • Narendra Karmarkar
  • William Karush
  • Leonid Khachiyan
  • Bernard Koopman
  • Harold Kuhn
  • Joseph Louis Lagrange
  • László Lovász
  • Arkadii Nemirovskii
  • Yurii Nesterov
  • John von Neumann
  • Boris Polyak
  • Lev Pontryagin
  • James Renegar
  • Kees Roos
  • Naum Z. Shor
  • Michael J. Todd
  • Albert Tucker

Major subfields

In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):

Multi-objective optimization

Adding more than one objective to an optimization problem adds complexity. For example, if you wanted to optimize a structural design, you would want a design that is both light and rigid. Because these two objectives conflict, a trade-off exists. There will be one lightest design, one stiffest design, and an infinite number of designs that are some compromise of weight and stiffness. This set of trade-off designs is known as a Pareto set. The curve created plotting weight against stiffness of the best designs is known as the Pareto frontier.

A design is judged to be Pareto optimal if it is not dominated by other designs: a Pareto optimal design must be better than another design in at least one aspect. If it is worse than another design in all respects, then it is dominated and is not Pareto optimal.

Multi-modal optimization

Optimization problems are often multi-modal, that is they possess multiple good solutions. They could all be globally good (same cost function value) or there could be a mix of globally good and locally good solutions. Obtaining all (or at least some of) the multiple solutions is the goal of a multi-modal optimizer.

Classical optimization techniques due to their iterative approach do not perform satisfactorily when they are used to obtain multiple solutions, since it is not guaranteed that different solutions will be obtained even with different starting points in multiple runs of the algorithm. Evolutionary Algorithms are however a very popular approach to obtain multiple solutions in a multi-modal optimization task. See Evolutionary multi-modal optimization.

Dimensionless optimization

Dimensionless optimization (DO) is used in design problems, and consists of the following steps:

Concepts and notation

Optimization problems

An optimization problem can be represented in the following way

Given: a function f : A \to R from some set A to the real numbers
Sought: an element x0 in A such that f(x0) ≤ f(x) for all x in A ("minimization") or such that f(x0) ≥ f(x) for all x in A ("maximization").

Such a formulation is called an optimization problem or a mathematical programming problem (a term not directly related to computer programming, but still in use for example in linear programming - see History above). Many real-world and theoretical problems may be modeled in this general framework. Problems formulated using this technique in the fields of physics and computer vision may refer to the technique as energy minimization, speaking of the value of the function f as representing the energy of the system being modeled.

Typically, A is some subset of the Euclidean space Rn, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy. The domain A of f is called the search space or the choice set, while the elements of A are called candidate solutions or feasible solutions.

The function f is called, variously, an objective function, cost function, energy function, or energy functional. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimal solution.

Generally, when the feasible region or the objective function of the problem does not present convexity, there may be several local minima and maxima, where a local minimum x* is defined as a point for which there exists some δ > 0 so that for all x such that

\|\mathbf{x}-\mathbf{x}^*\|\leq\delta;\,

the expression

f(\mathbf{x}^*)\leq f(\mathbf{x})

holds; that is to say, on some region around x* all of the function values are greater than or equal to the value at that point. Local maxima are defined similarly.

A large number of algorithms proposed for solving non-convex problems – including the majority of commercially available solvers – are not capable of making a distinction between local optimal solutions and rigorous optimal solutions, and will treat the former as actual solutions to the original problem. The branch of applied mathematics and numerical analysis that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a non-convex problem is called global optimization.

Notation

Optimization problems are often expressed with special notation. Here are some examples.

\min_{x\in\mathbb R}\; (x^2 + 1)

This asks for the minimum value for the objective function x2 + 1, where x ranges over the real numbers \mathbb R. The minimum value in this case is 1, occurring at x = 0.

\max_{x\in\mathbb R}\; 2x

This asks for the maximum value for the objective function 2x, where x ranges over the reals. In this case, there is no such maximum as the objective function is unbounded, so the answer is "infinity" or "undefined".

\operatorname{argmin}_{x\in[-\infty,-1]}\; x^2 + 1\,

This asks for the value (or values) of x in the interval [-\infty,-1] that minimizes (or minimize) the objective function x2 + 1 (the actual minimum value of that function does not matter). In this case, the answer is x = -1.

\operatorname{argmax}_{x\in[-5,5],\;y\in\mathbb R}\; x\cdot\cos(y)\,

This asks for the (x,y) pair (or pairs) that maximizes (or maximize) the value of the objective function x\cdot\cos(y), with the added constraint that x lies in the interval [-5,5] (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form (5, 2kπ) and (−5,(2k+1)π), where k ranges over all integers.

Analytical characterization of optima

Is it possible to satisfy all constraints?

The satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solution at all without regard to objective value. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal.

Many optimization algorithms need to start from a feasible point. One way to obtain such a point is to relax the feasibility conditions using a slack variable; with enough slack, any starting point is feasible. Then, minimize that slack variable until slack is null or negative.

Does an optimum exist?

The extreme value theorem of Karl Weierstrass states conditions under which an optimum exists.

How can an optimum be found?

One of Fermat's theorems states that optima of unconstrained problems are found at stationary points, where the first derivative or the gradient of the objective function is zero (see First derivative test). More generally, they may be found at critical points, where the first derivative or gradient of the objective function is zero or is undefined, or on the boundary of the choice set. An equation stating that the first derivative equals zero at an interior optimum is sometimes called a 'first-order condition'.

Optima of inequality-constrained problems are instead found by the Lagrange multiplier method. This method calculates a system of inequalities called the 'Karush-Kuhn-Tucker conditions' or 'complementary slackness conditions', which may then be used to calculate the optimum.

While the first derivative test identifies points that might be optima, it cannot distinguish a point which is a minimum from one that is a maximum or one that is neither. When the objective function is twice differentiable, these cases can be distinguished by checking the second derivative or the matrix of second derivatives (called the Hessian matrix) in unconstrained problems, or a matrix of second derivatives of the objective function and the constraints called the bordered Hessian. The conditions that distinguish maxima and minima from other stationary points are sometimes called 'second-order conditions' (see 'Second derivative test').

How does the optimum change if the problem changes?

The envelope theorem describes how the value of an optimal solution changes when an underlying parameter changes.

The maximum theorem of Claude Berge (1963) describes the continuity of the optimal solution as a function of underlying parameters.

Computational optimization techniques

Optimization methods are crudely divided into two groups:
SVO - Single-variable optimization
MVO - Multi-variable optimization

For twice-differentiable functions, unconstrained problems can be solved by finding the points where the gradient of the objective function is zero (that is, the stationary points) and using the Hessian matrix to classify the type of each point. If the Hessian is positive definite, the point is a local minimum, if negative definite, a local maximum, and if indefinite it is some kind of saddle point.

The existence of derivatives is not always assumed and many methods were devised for specific situations. The basic classes of methods, based on smoothness of the objective function, are:

Actual methods falling somewhere among the categories above include:

Should the objective function be convex over the region of interest, then any local minimum will also be a global minimum. There exist robust, fast numerical techniques for optimizing twice differentiable convex functions.

Constrained problems can often be transformed into unconstrained problems with the help of Lagrange multipliers.

Here are a few other popular methods:

Applications

Problems in rigid body dynamics (in particular articulated rigid body dynamics) often require mathematical programming techniques, since you can view rigid body dynamics as attempting to solve an ordinary differential equation on a constraint manifold; the constraints are various nonlinear geometric constraints such as "these two points must always coincide", "this surface must not penetrate any other", or "this point must always lie somewhere on this curve". Also, the problem of computing contact forces can be done by solving a linear complementarity problem, which can also be viewed as a QP (quadratic programming) problem.

Many design problems can also be expressed as optimization programs. This application is called design optimization. One subset is the engineering optimization, and another recent and growing subset of this field is multidisciplinary design optimization, which, while useful in many problems, has in particular been applied to aerospace engineering problems.

Economics also relies heavily on mathematical programming. An often studied problem in microeconomics, the utility maximization problem, and its dual problem the Expenditure minimization problem, are economic optimization problems. Consumers and firms are assumed to maximize their utility/profit. Also, agents are most frequently assumed to be risk-averse thereby wishing to minimize whatever risk they might be exposed to. Asset prices are also explained using optimization though the underlying theory is more complicated than simple utility or profit optimization. Trade theory also uses optimization to explain trade patterns between nations.

Another field that uses optimization techniques extensively is operations research.

See also

  • Curve fitting
  • Arg max
  • Brachistochrone
  • Fuzzy logic
  • Game theory
  • Goal programming
  • Important publications in optimization
  • Interval finite element
  • Least squares
  • Mathematical optimization software
  • Optimization algorithms
  • Optimization problem
  • Process optimization
  • Radial basis function
  • Random optimization
  • Topkis's theorem
  • Variational calculus

Solvers

References

  1. ↑ Merlin.cs.uoi.gr
  2. ↑ Acts.nersc.gov

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