Mathematical optimization

"Mathematical programming" redirects here. For the peer-reviewed journal, seeMathematical Programming.

"Optimization" and "Optimum" redirect here. For other uses, seeOptimization (disambiguation)andOptimum (disambiguation).

Mathematical optimization(alternatively spelledoptimisation) ormathematical programmingis 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 optimizationandcontinuous optimization.Optimization problems arise in all quantitative disciplines fromcomputer scienceandengineeringtooperations researchandeconomics,and the development of solution methods has been of interest inmathematicsfor centuries.

Graph of a surface given byz= f(x,y) = −(x² +y²) + 4. The globalmaximumat (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 ($f(x)$best) value.

In the more general approach, anoptimization problemconsists ofmaximizing or minimizingareal functionby systematically choosinginputvalues from within an allowed set and computing thevalueof the function. The generalization of optimization theory and techniques to other formulations constitutes a large area ofapplied mathematics.

Optimization problems

Main article:Optimization problem

Optimization problems can be divided into two categories, depending on whether thevariablesarecontinuousordiscrete:

• An optimization problem with discrete variables is known as adiscrete optimization,in which anobjectsuch as aninteger,permutationorgraphmust be found from acountable set.
• A problem with continuous variables is known as acontinuous optimization,in which optimal arguments from a continuous set must be found. They can includeconstrained problemsand multimodal problems.

An optimization problem can be represented in the following way:

Given:afunctionf:A→$\mathbb {R}$from somesetAto thereal numbers

Sought:an elementx₀∈Asuch thatf(x₀) ≤f(x)for allx∈A( "minimization" ) or such thatf(x₀) ≥f(x)for allx∈A( "maximization" ).

Such a formulation is called anoptimization problemor amathematical programming problem(a term not directly related tocomputer programming,but still in use for example inlinear programming– seeHistorybelow). 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 ofphysicsmay refer to the technique asenergyminimization,speaking of the value of the functionfas representing the energy of thesystembeingmodeled.Inmachine learning,it is always necessary to continuously evaluate the quality of a data model by using acost functionwhere a minimum implies a set of possibly optimal parameters with an optimal (lowest) error.

Typically,Ais somesubsetof theEuclidean space$\mathbb {R} ^{n}$,often specified by a set ofconstraints,equalities or inequalities that the members ofAhave to satisfy. ThedomainAoffis called thesearch spaceor thechoice set,while the elements ofAare calledcandidate solutionsorfeasible solutions.

The functionfis variously called anobjective function,criterion function,loss function,cost function(minimization),utility functionorfitness function(maximization), or, in certain fields, anenergy functionorenergyfunctional.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> 0such that

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

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 minimumis at least as good as every feasible element. Generally, unless the objective function isconvexin 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 optimizationis the branch ofapplied mathematicsandnumerical analysisthat 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.

Notation

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

Minimum and maximum value of a function

Consider the following notation:

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

This denotes the minimumvalueof the objective functionx²+ 1,when choosingxfrom the set ofreal numbers$\mathbb {R}$.The minimum value in this case is 1, occurring atx= 0.

Similarly, the notation

$\max _{x\in \mathbb {R} }\;2x$

asks for the maximum value of the objective function2x,wherexmay 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".

Optimal input arguments

Main article:Arg max

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 theargumentxin 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= 0is infeasible, that is, it does not belong to thefeasible 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 functionxcosy,with the added constraint thatxlie 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},wherekranges over allintegers.

Operatorsarg minandarg maxare sometimes also written asargminandargmax,and stand forargument of the minimumandargument of the maximum.

History

FermatandLagrangefound calculus-based formulae for identifying optima, whileNewtonandGaussproposed 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 Kantorovichin 1939. (Programmingin this context does not refer tocomputer programming,but comes from the use ofprogramby theUnited Statesmilitary to refer to proposed training andlogisticsschedules, which were the problems Dantzig studied at that time.) Dantzig published theSimplex algorithmin 1947, and alsoJohn von Neumannand other researchers worked on the theoretical aspects of linear programming (like the theory ofduality) 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

Major subfields

• Convex programmingstudies 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 functionfis linear and the constraints are specified using only linear equalities and inequalities. Such a constraint set is called apolyhedronor apolytopeif it isbounded.
  • 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 aresemidefinitematrices.It is a generalization of linear and convex quadratic programming.
  • Conic programmingis a general form of convex programming. LP, SOCP and SDP can all be viewed as conic programs with the appropriate type of cone.
  • Geometric programmingis a technique whereby objective and inequality constraints expressed asposynomialsand equality constraints asmonomialscan be transformed into a convex program.
• Integer programmingstudies linear programs in which some or all variables are constrained to take onintegervalues. This is not convex, and in general much more difficult than regular linear programming.
• Quadratic programmingallows 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 programmingstudies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem.
• Nonlinear programmingstudies 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 programmingstudies the case in which some of the constraints or parameters depend onrandom variables.
• Robust optimizationis, 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 optimizationis concerned with problems where the set of feasible solutions is discrete or can be reduced to adiscreteone.
• Stochastic optimizationis used with random (noisy) function measurements or random inputs in the search process.
• Infinite-dimensional optimizationstudies the case when the set of feasible solutions is a subset of an infinite-dimensionalspace, such as a space of functions.
• Heuristicsandmetaheuristicsmake 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 satisfactionstudies the case in which the objective functionfis constant (this is used inartificial intelligence,particularly inautomated reasoning).
  • Constraint programmingis 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 mappingis 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 variationsis 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 controltheory is a generalization of the calculus of variations which introduces control policies.
• Dynamic programmingis the approach to solve thestochastic optimizationproblem 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.
• Mathematical programming with equilibrium constraintsis where the constraints includevariational inequalitiesorcomplementarities.

Multi-objective optimization

Main article:Multi-objective optimization

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 optimizationproblems where the (partial) ordering is no longer given by the Pareto ordering.

Multi-modal or global 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.

Common approaches toglobal optimizationproblems, where multiple local extrema may be present includeevolutionary algorithms,Bayesian optimizationandsimulated annealing.

Classification of critical points and extrema

Feasibility problem

Thesatisfiability problem,also called thefeasibility problem,is just the problem of finding anyfeasible solutionat 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 torelaxthe 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.

Existence

Theextreme value theoremofKarl Weierstrassstates 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.

Necessary conditions for optimality

One of Fermat's theoremsstates 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 multipliermethod. The optima of problems with equality and/or inequality constraints can be found using the 'Karush–Kuhn–Tucker conditions'.

Sufficient conditions for optimality

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 Hessianin 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.

Sensitivity and continuity of optima

Theenvelope theoremdescribes how the value of an optimal solution changes when an underlyingparameterchanges. The process of computing this change is calledcomparative statics.

Themaximum theoremofClaude Berge(1963) describes the continuity of an optimal solution as a function of underlying parameters.

Calculus of optimization

Main article:Karush–Kuhn–Tucker conditions

See also:Critical point (mathematics),Differential calculus,Gradient,Hessian matrix,Definite matrix,Lipschitz continuity,Rademacher's theorem,Convex function,andConvex analysis

For unconstrained problems with twice-differentiable functions, somecritical pointscan be found by finding the points where thegradientof the objective function is zero (that is, the stationary points). More generally, a zerosubgradientcertifies that a local minimum has been found forminimization problems with convexfunctionsand 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.

Further, critical points can be classified using thedefinitenessof theHessian matrix:If the Hessian ispositivedefinite 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 relaxationcan 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.

Global convergence

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.

Computational optimization techniques

To solve problems, researchers may usealgorithmsthat terminate in a finite number of steps, oriterative methodsthat converge to a solution (on some specified class of problems), orheuristicsthat may provide approximate solutions to some problems (although their iterates need not converge).

Optimization algorithms

For a more comprehensive list, seeList of optimization algorithms.

• Simplex algorithmofGeorge Dantzig,designed forlinear programming
• Extensions of the simplex algorithm, designed forquadratic programmingand forlinear-fractional programming
• Variants of the simplex algorithm that are especially suited fornetwork optimization
• Combinatorial algorithms
• Quantum optimization algorithms

Iterative methods

Main article:Iterative method

Theiterative methodsused to solve problems ofnonlinear programmingdiffer 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):
  • Newton's method
  • Sequential quadratic programming:A Newton-based method for small-medium scaleconstrainedproblems. 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 descentmethods: Algorithms which update a single coordinate in each iteration
  • Conjugate gradient methods:Iterative methodsfor 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.
  • Subgradient methods:An iterative method for largelocallyLipschitz functionsusinggeneralized gradients.Following Boris T. Polyak, subgradient–projection methods are similar to conjugate–gradient methods.
  • Bundle method of descent: An iterative method for small–medium-sized problems with locally Lipschitz functions, particularly forconvex minimizationproblems (similar to conjugate gradient methods).
  • Ellipsoid method:An iterative method for small problems withquasiconvexobjective 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).
  • Simultaneous perturbation stochastic approximation(SPSA) method for stochastic optimization; uses random (efficient) gradient approximation.
• 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.
  • Interpolationmethods
  • Pattern searchmethods, which have better convergence properties than theNelder–Mead heuristic (with simplices),which is listed below.
  • Mirror descent

Heuristics

Main article:Heuristic algorithm

Besides (finitely terminating)algorithmsand (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:

• Differential evolution
• Dynamic relaxation
• Evolutionary algorithms
• Genetic algorithms
• Hill climbingwith random restart
• Memetic algorithm
• Nelder–Mead simplicial heuristic:A popular heuristic for approximate minimization (without calling gradients)
• Particle swarm optimization
• Simulated annealing
• Stochastic tunneling
• Tabu search

Applications

Mechanics

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 equationon 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 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 engineeringproblems.

This approach may be applied in cosmology and astrophysics.

Economics and finance

Economicsis closely enough linked to optimization ofagentsthat an influential definition relatedly describes economicsquascience as the "study of human behavior as a relationship between ends andscarcemeans "with alternative uses.Modern optimization theory includes traditional optimization theory but also overlaps withgame theoryand the study of economicequilibria.TheJournal of Economic Literaturecodesclassify mathematical programming, optimization techniques, and related topics underJEL:C61-C63.

In microeconomics, theutility maximization problemand itsdual problem,theexpenditure minimization problem,are economic optimization problems. Insofar as they behave consistently,consumersare assumed to maximize theirutility,whilefirmsare usually assumed to maximize theirprofit.Also, agents are often modeled as beingrisk-averse,thereby preferring to avoid risk.Asset pricesare also modeled using optimization theory, though the underlying mathematics relies on optimizingstochastic processesrather than on static optimization.International trade theoryalso uses optimization to explain trade patterns between nations. The optimization ofportfoliosis an example of multi-objective optimization in economics.

Since the 1970s, economists have modeled dynamic decisions over time usingcontrol theory.For example, dynamicsearch modelsare used to studylabor-market behavior.A crucial distinction is between deterministic and stochastic models.Macroeconomistsbuilddynamic stochastic general equilibrium (DSGE)models that describe the dynamics of the whole economy as the result of the interdependent optimizing decisions of workers, consumers, investors, and governments.

Electrical engineering

Some common applications of optimization techniques inelectrical engineeringincludeactive filterdesign,stray field reduction in superconducting magnetic energy storage systems,space mappingdesign ofmicrowavestructures,handset antennas,electromagnetics-based design. Electromagnetically validated design optimization of microwave components and antennas has made extensive use of an appropriate physics-based or empiricalsurrogate modelandspace mappingmethodologies since the discovery ofspace mappingin 1993.Optimization techniques are also used inpower-flow analysis.

Civil engineering

Optimization has been widely used in civil engineering.Construction managementandtransportation engineeringare among the main branches of civil engineering that heavily rely on optimization. The most common civil engineering problems that are solved by optimization are cut and fill of roads, life-cycle analysis of structures and infrastructures,resource leveling,water resource allocation,trafficmanagementand schedule optimization.

Operations research

Another field that uses optimization techniques extensively isoperations research.Operations research also uses stochastic modeling and simulation to support improved decision-making. Increasingly, operations research usesstochastic programmingto model dynamic decisions that adapt to events; such problems can be solved with large-scale optimization andstochastic optimizationmethods.

Control engineering

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.

Geophysics

Optimization techniques are regularly used ingeophysicalparameter estimation problems. Given a set of geophysical measurements, e.g.seismic recordings,it is common to solve for thephysical propertiesandgeometrical shapesof the underlying rocks and fluids. The majority of problems in geophysics are nonlinear with both deterministic and stochastic methods being widely used.

Molecular modeling

Main article:Molecular modeling

Nonlinear optimization methods are widely used inconformational analysis.

Computational systems biology

Main article:Computational systems biology

Optimization techniques are used in many facets of computational systems biology such as model building, optimal experimental design, metabolic engineering, and synthetic biology.Linear programminghas been applied to calculate the maximal possible yields of fermentation products,and to infer gene regulatory networks from multiple microarray datasetsas well as transcriptional regulatory networks from high-throughput data.Nonlinear programminghas been used to analyze energy metabolismand has been applied to metabolic engineering and parameter estimation in biochemical pathways.

Machine learning

Main article:Machine learning § Optimization

Solvers

Main article:List of optimization software

See also

• Brachistochrone curve
• Curve fitting
• Deterministic global optimization
• Goal programming
• Important publications in optimization
• Least squares
• Mathematical Optimization Society(formerly Mathematical Programming Society)
• Mathematical optimization algorithms
• Mathematical optimization software
• Process optimization
• Simulation-based optimization
• Test functions for optimization
• Vehicle routing problem

Further reading

• Boyd, Stephen P.;Vandenberghe, Lieven (2004).Convex Optimization.Cambridge: Cambridge University Press.ISBN0-521-83378-7.
• Gill, P. E.; Murray, W.;Wright, M. H.(1982).Practical Optimization.London: Academic Press.ISBN0-12-283952-8.
• Lee, Jon(2004).A First Course in Combinatorial Optimization.Cambridge University Press.ISBN0-521-01012-8.
• Nocedal, Jorge;Wright, Stephen J. (2006).Numerical Optimization(2nd ed.). Berlin: Springer.ISBN0-387-30303-0.
• G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd (eds.):Optimization,Elsevier, (1989).
• Stanislav Walukiewicz:Integer Programming,Springer, ISBN 978-9048140688, (1990).
• R. Fletcher:Practical Methods of Optimization,2nd Ed.,Wiley, (2000).
• Panos M. Pardalos:Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems,Springer, ISBN 978-1-44194829-8, (2000).
• Xiaoqi Yang, K. L. Teo, Lou Caccetta (Eds.):Optimization Methods and Applications,Springer, ISBN 978-0-79236866-3, (2001).
• Panos M. Pardalos, and Mauricio G. C. Resende(Eds.):Handbook of Applied Optimization,Oxford Univ Pr on Demand, ISBN 978-0-19512594-8, (2002).
• Wil Michiels, Emile Aarts, and Jan Korst:Theoretical Aspects of Local Search,Springer, ISBN 978-3-64207148-5, (2006).
• Der-San Chen, Robert G. Batson,and Yu Dang:Applied Integer Programming: Modeling and Solution,Wiley, ISBN 978-0-47037306-4, (2010).
• Mykel J. Kochenderfer and Tim A. Wheeler:Algorithms for Optimization,The MIT Press, ISBN 978-0-26203942-0, (2019).
• 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).

External links

• "Decision Tree for Optimization Software".Links to optimization source codes
• "Global optimization".
• "EE364a: Convex Optimization I".Course from Stanford University.
• Varoquaux, Gaël."Mathematical Optimization: Finding Minima of Functions".