Graph of a surface given byz = f(x,y) = −(x² +y²) + 4. The globalmaximum at (x, y, z) = (0, 0, 4) is indicated by a blue dot.Nelder-Mead minimum search ofSimionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value.
Mathematical optimization (alternatively spelledoptimisation) ormathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives.[1][2] It is generally divided into two subfields:discrete optimization andcontinuous optimization. Optimization problems arise in all quantitative disciplines fromcomputer science andengineering[3] tooperations research andeconomics, and the development of solution methods has been of interest inmathematics for centuries.[4]
A problem with continuous variables is known as acontinuous optimization, in which optimal arguments from a continuous set must be found. They can includeconstrained problems and multimodal problems.
An optimization problem can be represented in the following way:
Sought: an elementx0 ∈A such thatf(x0) ≤f(x) for allx ∈A ("minimization") or such thatf(x0) ≥f(x) for allx ∈A ("maximization").
Such a formulation is called anoptimization problem or amathematical programming problem (a term not directly related tocomputer programming, but still in use for example inlinear programming – seeHistory below). Many real-world and theoretical problems may be modeled in this general framework.
Since the following is valid:
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 ofphysics may refer to the technique asenergy minimization,[5] speaking of the value of the functionf as representing the energy of thesystem beingmodeled. Inmachine learning, it is always necessary to continuously evaluate the quality of a data model by using acost function where a minimum implies a set of possibly optimal parameters with an optimal (lowest) error.
Typically,A is somesubset of theEuclidean space, often specified by a set ofconstraints, equalities or inequalities that the members ofA have to satisfy. ThedomainA off is called thesearch space or thechoice set, while the elements ofA are calledcandidate solutions orfeasible solutions.
The functionf is variously called anobjective function,criterion function,loss function,cost function (minimization),[6]utility function orfitness function (maximization), or, in certain fields, anenergy function orenergyfunctional. A feasible solution that minimizes (or maximizes) the objective function is called anoptimal solution.
In mathematics, conventional optimization problems are usually stated in terms of minimization.
Alocal minimumx* is defined as an element for which there exists someδ > 0 such that
the expressionf(x*) ≤f(x) holds;
that is to say, on some region aroundx* all of the function values are greater than or equal to the value at that element. Local maxima are defined similarly.
While a local minimum is at least as good as any nearby elements, aglobal minimum is at least as good as every feasible element.Generally, unless the objective function isconvex in a minimization problem, there may be several local minima.In aconvex problem, if there is a local minimum that is interior (not on the edge of the set of feasible elements), it is also the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima.
A large number of algorithms proposed for solving the nonconvex problems – including the majority of commercially available solvers – are not capable of making a distinction between locally optimal solutions and globally optimal solutions, and will treat the former as actual solutions to the original problem.Global optimization is the branch ofapplied mathematics andnumerical 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 nonconvex problem.
This denotes the minimumvalue of the objective functionx2 + 1, when choosingx from the set ofreal numbers. The minimum value in this case is 1, occurring atx = 0.
Similarly, the notation
asks for the maximum value of the objective function2x, wherex 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".
This represents the value (or values) of theargumentx in theinterval(−∞,−1] that minimizes (or minimize) the objective functionx2 + 1 (the actual minimum value of that function is not what the problem asks for). In this case, the answer isx = −1, sincex = 0 is infeasible, that is, it does not belong to thefeasible set.
Similarly,
or equivalently
represents the{x,y} pair (or pairs) that maximizes (or maximize) the value of the objective functionx cosy, with the added constraint thatx 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}, wherek ranges over allintegers.
Operatorsarg min andarg max are sometimes also written asargmin andargmax, and stand forargument of the minimum andargument of the maximum.
Fermat andLagrange found calculus-based formulae for identifying optima, whileNewton andGauss proposed iterative methods for moving towards an optimum.
The term "linear programming" for certain optimization cases was due toGeorge B. Dantzig, although much of the theory had been introduced byLeonid Kantorovich in 1939. (Programming in this context does not refer tocomputer programming, but comes from the use ofprogram by theUnited States military to refer to proposed training andlogistics schedules, which were the problems Dantzig studied at that time.) Dantzig published theSimplex algorithm in 1947, and alsoJohn von Neumann and other researchers worked on the theoretical aspects of linear programming (like the theory ofduality) around the same time.[7]
Other notable researchers in mathematical optimization include the following:
Convex programming studies the case when the objective function isconvex (minimization) orconcave (maximization) and the constraint set isconvex. 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 functionf is linear and the constraints are specified using only linear equalities and inequalities. Such a constraint set is called apolyhedron or apolytope if it isbounded.
Semidefinite programming (SDP) is a subfield of convex optimization where the underlying variables aresemidefinitematrices. 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 asposynomials and equality constraints asmonomials can be transformed into a convex program.
Integer programming studies linear programs in which some or all variables are constrained to take oninteger 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.
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 adiscrete one.
Stochastic optimization is used with random (noisy) function measurements or random inputs in the search process.
Heuristics andmetaheuristics 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 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 orsurrogate model.
In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):
Calculus of variations is concerned with finding the best way to achieve some goal, such as finding a surface whose boundary is a specific curve, but with the least possible area.
Optimal control theory is a generalization of the calculus of variations which introduces control policies.
Dynamic programming is the approach to solve thestochastic optimization problem with stochastic, randomness, and unknown model parameters. It studies the case in which the optimization strategy is based on splitting the problem into smaller subproblems. The equation that describes the relationship between these subproblems is called theBellman equation.
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 thePareto set. The curve created plotting weight against stiffness of the best designs is known as thePareto 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 intovector 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.
Thesatisfiability problem, also called thefeasibility problem, is just the problem of finding anyfeasible 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 torelax the feasibility conditions using aslack variable; with enough slack, any starting point is feasible. Then, minimize that slack variable until the slack is null or negative.
Theextreme value theorem ofKarl 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 atstationary points, where the first derivative or the gradient of the objective function is zero (seefirst derivative test). More generally, they may be found atcritical 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 theLagrange 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 theHessian matrix) in unconstrained problems, or the matrix of second derivatives of the objective function and the constraints called thebordered 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.
Theenvelope theorem describes how the value of an optimal solution changes when an underlyingparameter changes. The process of computing this change is calledcomparative statics.
Themaximum theorem ofClaude Berge (1963) describes the continuity of an optimal solution as a function of underlying parameters.
For unconstrained problems with twice-differentiable functions, somecritical points can be found by finding the points where thegradient of the objective function is zero (that is, the stationary points). More generally, a zerosubgradient certifies that a local minimum has been found forminimization problems with convexfunctions and otherlocallyLipschitz 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.[8]
Further, critical points can be classified using thedefiniteness of theHessian matrix: If the Hessian ispositive 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 ofsaddle point.
Constrained problems can often be transformed into unconstrained problems with the help ofLagrange multipliers.Lagrangian relaxation can also provide approximate solutions to difficult constrained problems.
When the objective function is aconvex function, then any local minimum will also be a global minimum. There exist efficient numerical techniques for minimizing convex functions, such asinterior-point methods.
More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies online searches, which optimize a function along one dimension. A second and increasingly popular method for ensuring convergence usestrust regions. Both line searches and trust regions are used in modern methods ofnon-differentiable optimization. Usually, a global optimizer is much slower than advanced local optimizers (such asBFGS), so often an efficient global optimizer can be constructed by starting the local optimizer from different starting points.
To solve problems, researchers may usealgorithms that terminate in a finite number of steps, oriterative methods that converge to a solution (on some specified class of problems), orheuristics that may provide approximate solutions to some problems (although their iterates need not converge).
Theiterative methods used to solve problems ofnonlinear programming differ according to whether theyevaluateHessians, gradients, or only function values. While evaluating Hessians (H) and gradients (G) improves the rate of convergence, for functions for which these quantities exist and vary sufficiently smoothly, such evaluations increase thecomputational complexity (or computational cost) of each iteration. In some cases, the computational complexity may be excessively high.
One major criterion for optimizers is just the number of required function evaluations as this often is already a large computational effort, usually much more effort than within the optimizer itself, which mainly has to operate over the N variables. The derivatives provide detailed information for such optimizers, but are even harder to calculate, e.g. approximating the gradient takes at least N+1 function evaluations. For approximations of the 2nd derivatives (collected in the Hessian matrix), the number of function evaluations is in the order of N². Newton's method requires the 2nd-order derivatives, so for each iteration, the number of function calls is in the order of N², but for a simpler pure gradient optimizer it is only N. However, gradient optimizers need usually more iterations than Newton's algorithm. Which one is best with respect to the number of function calls depends on the problem itself.
Methods that evaluate Hessians (or approximate Hessians, usingfinite differences):
Sequential quadratic programming: A Newton-based method for small-medium scaleconstrained problems. Some versions can handle large-dimensional problems.
Interior point methods: This is a large class of methods for constrained optimization, some of which use only (sub)gradient information and others of which require the evaluation of Hessians.
Methods that evaluate gradients, or approximate gradients in some way (or even subgradients):
Coordinate descent methods: Algorithms which update a single coordinate in each iteration
Conjugate gradient methods:Iterative methods for large problems. (In theory, these methods terminate in a finite number of steps with quadratic objective functions, but this finite termination is not observed in practice on finite–precision computers.)
Gradient descent (alternatively, "steepest descent" or "steepest ascent"): A (slow) method of historical and theoretical interest, which has had renewed interest for finding approximate solutions of enormous problems.
Bundle method of descent: An iterative method for small–medium-sized problems with locally Lipschitz functions, particularly forconvex minimization problems (similar to conjugate gradient methods).
Ellipsoid method: An iterative method for small problems withquasiconvex objective functions and of great theoretical interest, particularly in establishing the polynomial time complexity of some combinatorial optimization problems. It has similarities with Quasi-Newton methods.
Conditional gradient method (Frank–Wolfe) for approximate minimization of specially structured problems withlinear constraints, especially with traffic networks. For general unconstrained problems, this method reduces to the gradient method, which is regarded as obsolete (for almost all problems).
Quasi-Newton methods: Iterative methods for medium-large problems (e.g. N<1000).
Methods that evaluate only function values: If a problem is continuously differentiable, then gradients can be approximated using finite differences, in which case a gradient-based method can be used.
Besides (finitely terminating)algorithms and (convergent)iterative methods, there areheuristics. A heuristic is any algorithm which is not guaranteed (mathematically) to find the solution, but which is nevertheless useful in certain practical situations. List of some well-known heuristics:
Problems inrigid body dynamics (in particular articulated rigid body dynamics) often require mathematical programming techniques, since you can view rigid body dynamics as attempting to solve anordinary differential equation on a constraint manifold;[9] 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 alinear 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 theengineering optimization, and another recent and growing subset of this field ismultidisciplinary design optimization, which, while useful in many problems, has in particular been applied toaerospace engineering problems.
This approach may be applied in cosmology and astrophysics.[10]
Economics is closely enough linked to optimization ofagents that an influential definition relatedly describes economicsqua science as the "study of human behavior as a relationship between ends andscarce means" with alternative uses.[11] Modern optimization theory includes traditional optimization theory but also overlaps withgame theory and the study of economicequilibria. TheJournal of Economic Literaturecodes classify mathematical programming, optimization techniques, and related topics underJEL:C61-C63.
In microeconomics, theutility maximization problem and itsdual problem, theexpenditure minimization problem, are economic optimization problems. Insofar as they behave consistently,consumers are assumed to maximize theirutility, whilefirms are usually assumed to maximize theirprofit. Also, agents are often modeled as beingrisk-averse, thereby preferring to avoid risk.Asset prices are also modeled using optimization theory, though the underlying mathematics relies on optimizingstochastic processes rather than on static optimization.International trade theory also uses optimization to explain trade patterns between nations. The optimization ofportfolios is an example of multi-objective optimization in economics.
Some common applications of optimization techniques inelectrical engineering includeactive filter design,[17] stray field reduction in superconducting magnetic energy storage systems,space mapping design ofmicrowave structures,[18] handset antennas,[19][20][21] electromagnetics-based design. Electromagnetically validated design optimization of microwave components and antennas has made extensive use of an appropriate physics-based or empiricalsurrogate model andspace mapping methodologies since the discovery ofspace mapping in 1993.[22][23] Optimization techniques are also used inpower-flow analysis.[24]
Another field that uses optimization techniques extensively isoperations research.[29] Operations research also uses stochastic modeling and simulation to support improved decision-making. Increasingly, operations research usesstochastic programming to model dynamic decisions that adapt to events; such problems can be solved with large-scale optimization andstochastic optimization methods.
Mathematical optimization is used in much modern controller design. High-level controllers such asmodel predictive control (MPC) or real-time optimization (RTO) employ mathematical optimization. These algorithms run online and repeatedly determine values for decision variables, such as choke openings in a process plant, by iteratively solving a mathematical optimization problem including constraints and a model of the system to be controlled.
Optimization techniques are regularly used ingeophysical parameter estimation problems. Given a set of geophysical measurements, e.g.seismic recordings, it is common to solve for thephysical properties andgeometrical shapes of the underlying rocks and fluids. The majority of problems in geophysics are nonlinear with both deterministic and stochastic methods being widely used.
Optimization techniques are used in many facets of computational systems biology such as model building, optimal experimental design, metabolic engineering, and synthetic biology.[30]Linear programming has been applied to calculate the maximal possible yields of fermentation products,[30] and to infer gene regulatory networks from multiple microarray datasets[31] as well as transcriptional regulatory networks from high-throughput data.[32]Nonlinear programming has been used to analyze energy metabolism[33] and has been applied to metabolic engineering and parameter estimation in biochemical pathways.[34]
^Du, D. Z.; Pardalos, P. M.; Wu, W. (2008). "History of Optimization". InFloudas, C.; Pardalos, P. (eds.).Encyclopedia of Optimization. Boston: Springer. pp. 1538–1542.
^Hartmann, Alexander K; Rieger, Heiko (2002).Optimization algorithms in physics. Citeseer.
^De, Bishnu Prasad; Kar, R.; Mandal, D.; Ghoshal, S.P. (2014-09-27). "Optimal selection of components value for analog active filter design using simplex particle swarm optimization".International Journal of Machine Learning and Cybernetics.6 (4):621–636.doi:10.1007/s13042-014-0299-0.ISSN1868-8071.S2CID13071135.
^Bandler, J.W.; Biernacki, R.M.; Chen, Shao Hua; Grobelny, P.A.; Hemmers, R.H. (1994). "Space mapping technique for electromagnetic optimization".IEEE Transactions on Microwave Theory and Techniques.42 (12):2536–2544.Bibcode:1994ITMTT..42.2536B.doi:10.1109/22.339794.
^Bandler, J.W.; Biernacki, R.M.; Shao Hua Chen; Hemmers, R.H.; Madsen, K. (1995). "Electromagnetic optimization exploiting aggressive space mapping".IEEE Transactions on Microwave Theory and Techniques.43 (12):2874–2882.Bibcode:1995ITMTT..43.2874B.doi:10.1109/22.475649.
^Convex relaxation of optimal power flow: A tutorial. 2013 iREP Symposium on Bulk Power System Dynamics and Control.doi:10.1109/IREP.2013.6629391.
^Hegazy, Tarek (June 1999). "Optimization of Resource Allocation and Leveling Using Genetic Algorithms".Journal of Construction Engineering and Management.125 (3):167–175.doi:10.1061/(ASCE)0733-9364(1999)125:3(167).
^Piryonesi, S. Madeh; Nasseri, Mehran; Ramezani, Abdollah (9 July 2018). "Piryonesi, S. M., Nasseri, M., & Ramezani, A. (2018). Resource leveling in construction projects with activity splitting and resource constraints: a simulated annealing optimization".Canadian Journal of Civil Engineering.46:81–86.doi:10.1139/cjce-2017-0670.hdl:1807/93364.S2CID116480238.
^Vo, Thuy D.; Paul Lee, W.N.; Palsson, Bernhard O. (May 2007). "Systems analysis of energy metabolism elucidates the affected respiratory chain complex in Leigh's syndrome".Molecular Genetics and Metabolism.91 (1):15–22.doi:10.1016/j.ymgme.2007.01.012.ISSN1096-7192.PMID17336115.
Vladislav Bukshtynov:Optimization: Success in Practice, CRC Press (Taylor & Francis), ISBN 978-1-03222947-8, (2023) .
Rosario Toscano:Solving Optimization Problems with the Heuristic Kalman Algorithm: New Stochastic Methods, Springer, ISBN 978-3-031-52458-5 (2024).
Immanuel M. Bomze, Tibor Csendes, Reiner Horst and Panos M. Pardalos:Developments in Global Optimization, Kluwer Academic, ISBN 978-1-4419-4768-0 (2010).
