Optimization (mathematics)

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.

In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics.

Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete:

• An optimization problem with discrete variables is known as a discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set.
• A problem with continuous variables is known as a continuous optimization, in which optimal arguments from a continuous set must be found. They can include constrained problems and multimodal problems.

An optimization problem can be represented in the following way:

Given: a function \(f:A \rarr \mathbb R\) from some set A to the real numbers

Sought: an element x₀ ∈ A such that f(x₀) ≤ f(x) for all x ∈ A ("minimization") or such that f(x₀) ≥ f(x) for all x ∈ 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 below). Many real-world and theoretical problems may be modeled in this general framework.

Since the following is valid:

\(f(\mathbf{x}_{0})\geq f(\mathbf{x}) \Leftrightarrow -f(\mathbf{x}_{0})\leq -f(\mathbf{x}),\)

it suffices to solve only minimization problems. However, the opposite perspective of considering only maximization problems would be valid, too.

Problems formulated using this technique in the fields of physics 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. In machine learning, it is always necessary to continuously evaluate the quality of a data model by using a cost function where a minimum implies a set of possibly optimal parameters with an optimal (lowest) error.

Typically, A is some subset of the Euclidean space \(\mathbb R^n\), 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.

\(\forall\mathbf{x}\in A \; \text{where} \;\left\Vert\mathbf{x}-\mathbf{x}^{\ast}\right\Vert\leq\delta,\,\)

Condensed: the full section is in Wikipedia.

Consider the following notation:

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

This denotes the minimum value of the objective function x + 1, when choosing x from the set of real numbers \(\mathbb{R}\). The minimum value in this case is 1, occurring at x = 0.

Similarly, the notation

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

asks for the maximum value of the objective function 2x, where x may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "infinity" or "undefined".

Consider the following notation:

\(\underset{x\in(-\infty,-1]}{\operatorname{arg\,min}} \; x^2 + 1,\)

or equivalently

\(\underset{x}{\operatorname{arg\,min}} \; x^2 + 1, \; \text{subject to:} \; x\in(-\infty,-1].\)

This represents the value (or values) of the argument x in the interval (−∞,−1] that minimizes (or minimize) the objective function x + 1 (the actual minimum value of that function is not what the problem asks for). In this case, the answer is x = −1, since x = 0 is infeasible, that is, it does not belong to the feasible set.

Similarly,

\(\underset{x\in[-5,5], \; y\in\mathbb R}{\operatorname{arg\,max}} \; x\cos y,\)

or equivalently

\(\underset{x, \; y}{\operatorname{arg\,max}} \; x\cos y, \; \text{subject to:} \; x\in[-5,5], \; y\in\mathbb R,\)

represents the {x, y} pair (or pairs) that maximizes (or maximize) the value of the objective function x cos y, with the added constraint that x lie 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.

Operators arg min and arg max are sometimes also written as argmin and argmax, and stand for argument of the minimum and argument of the maximum.

Fermat and Lagrange found calculus-based formulae for identifying optima, while Newton and Gauss proposed iterative methods for moving towards an optimum.

The term "linear programming" for certain optimization cases was due to George B. Dantzig, although much of the theory had been introduced by Leonid Kantorovich in 1939. (Programming in this context does not refer to computer programming, but comes from the use of program by the United States military to refer to proposed training and logistics schedules, which were the problems Dantzig studied at that time.) Dantzig published the Simplex algorithm in 1947, and also John von Neumann and other researchers worked on the theoretical aspects of linear programming (like the theory of duality) around the same time.

Other notable researchers in mathematical optimization include the following:

• Richard Bellman
• Dimitri Bertsekas
• Michel Bierlaire
• Stephen P. Boyd
• Roger Fletcher
• Martin Grötschel
• Ronald A. Howard
• Fritz John
• Narendra Karmarkar
• William Karush
• Leonid Khachiyan
• Bernard Koopman
• Harold Kuhn
• László Lovász
• David Luenberger
• Arkadi Nemirovski
• Yurii Nesterov
• Lev Pontryagin
• R. Tyrrell Rockafellar
• Naum Z. Shor
• Albert Tucker

• Convex programming studies the case when the objective function is convex (minimization) or concave (maximization) and the constraint set is convex. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.
  • Linear programming (LP), a type of convex programming, studies the case in which the objective function f is linear and the constraints are specified using only linear equalities and inequalities. Such a constraint set is called a polyhedron or a polytope if it is bounded.
  • Second-order cone programming (SOCP) is a convex program, and includes certain types of quadratic programs.
  • Semidefinite programming (SDP) is a subfield of convex optimization where the underlying variables are semidefinite matrices. It is a generalization of linear and convex quadratic programming.
  • Conic programming is a general form of convex programming. LP, SOCP and SDP can all be viewed as conic programs with the appropriate type of cone.
  • Geometric programming is a technique whereby objective and inequality constraints expressed as posynomials and equality constraints as monomials can be transformed into a convex program.
• Integer programming studies linear programs in which some or all variables are constrained to take on integer values. This is not convex, and in general much more difficult than regular linear programming.
• Quadratic programming allows the objective function to have quadratic terms, while the feasible set must be specified with linear equalities and inequalities. For specific forms of the quadratic term, this is a type of convex programming.
• Fractional programming studies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem.
• Nonlinear programming studies the general case in which the objective function or the constraints or both contain nonlinear parts. This may or may not be a convex program. In general, whether the program is convex affects the difficulty of solving it.
• Stochastic programming studies the case in which some of the constraints or parameters depend on random variables.
• Robust optimization is, like stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem. Robust optimization aims to find solutions that are valid under all possible realizations of the uncertainties defined by an uncertainty set.
• Combinatorial optimization is concerned with problems where the set of feasible solutions is discrete or can be reduced to a discrete one.
• Stochastic optimization is used with random (noisy) function measurements or random inputs in the search process.
• Infinite-dimensional optimization studies the case when the set of feasible solutions is a subset of an infinite-dimensional space, such as a space of functions.
• Heuristics and metaheuristics make few or no assumptions about the problem being optimized. Usually, heuristics do not guarantee that any optimal solution need be found. On the other hand, heuristics are used to find approximate solutions for many complicated optimization problems.
• Constraint satisfaction studies the case in which the objective function f is constant (this is used in artificial intelligence, particularly in automated reasoning).
  • Constraint programming is a programming paradigm wherein relations between variables are stated in the form of constraints.
• Disjunctive programming is used where at least one constraint must be satisfied but not all. It is of particular use in scheduling.
• Space mapping is a concept for modeling and optimization of an engineering system to high-fidelity (fine) model accuracy exploiting a suitable physically meaningful coarse or surrogate model.

Condensed: the full section is in Wikipedia.

Adding more than one objective to an optimization problem adds complexity. For example, to optimize a structural design, one would desire a design that is both light and rigid. When two objectives conflict, a trade-off must be created. There may be one lightest design, one stiffest design, and an infinite number of designs that are some compromise of weight and rigidity. The set of trade-off designs that improve upon one criterion at the expense of another is known as the 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" (equivalently, "Pareto efficient" or in the Pareto set) if it is not dominated by any other design: If it is worse than another design in some respects and no better in any respect, then it is dominated and is not Pareto optimal.

The choice among "Pareto optimal" solutions to determine the "favorite solution" is delegated to the decision maker. In other words, defining the problem as multi-objective optimization signals that some information is missing: desirable objectives are given but combinations of them are not rated relative to each other. In some cases, the missing information can be derived by interactive sessions with the decision maker.

Multi-objective optimization problems have been generalized further into vector optimization problems where the (partial) ordering is no longer given by the Pareto ordering.

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.

Common approaches to global optimization problems, where multiple local extrema may be present include evolutionary algorithms, Bayesian optimization and simulated annealing.

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 the slack is null or negative.

The extreme value theorem of Karl Weierstrass states that a continuous real-valued function on a compact set attains its maximum and minimum value. More generally, a lower semi-continuous function on a compact set attains its minimum; an upper semi-continuous function on a compact set attains its maximum point or view.

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 (or set of equations) stating that the first derivative(s) equal(s) zero at an interior optimum is called a 'first-order condition' or a set of first-order conditions.

Optima of equality-constrained problems can be found by the Lagrange multiplier method. The optima of problems with equality and/or inequality constraints can be found using the 'Karush-Kuhn-Tucker conditions'.

While the first derivative test identifies points that might be extrema, this test does not distinguish a point that 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 the matrix of second derivatives of the objective function and the constraints called the bordered Hessian in constrained problems. The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' (see 'Second derivative test'). If a candidate solution satisfies the first-order conditions, then the satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.

The envelope theorem describes how the value of an optimal solution changes when an underlying parameter changes. The process of computing this change is called comparative statics.

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

For unconstrained problems with twice-differentiable functions, some critical points can be found by finding the points where the gradient of the objective function is zero (that is, the stationary points). More generally, a zero subgradient certifies that a local minimum has been found for minimization problems with convex functions and other locally Lipschitz functions, which meet in loss function minimization of the neural network. The positive-negative momentum estimation lets to avoid the local minimum and converges at the objective function global minimum.

Further, critical points can be classified using the definiteness of the Hessian matrix: If the Hessian is positive definite at a critical point, then the point is a local minimum; if the Hessian matrix is negative definite, then the point is a local maximum; finally, if indefinite, then the point is some kind of saddle point.

Constrained problems can often be transformed into unconstrained problems with the help of Lagrange multipliers. Lagrangian relaxation can also provide approximate solutions to difficult constrained problems.

When the objective function is a convex function, then any local minimum will also be a global minimum. There exist efficient numerical techniques for minimizing convex functions, such as interior-point methods.