In mathematics, computer science and operations research, mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element (with regard to some criterion) from some set of available alternatives.
In the simplest case, 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. More generally, optimization includes finding "best available" values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains.
An optimization problem can be represented in the following way:
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. 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 as representing the energy of the system being modeled.
Typically, is some subset of the Euclidean space , often specified by a set of constraints, equalities or inequalities that the members of have to satisfy. The domain of is called the search space or the choice set, while the elements of are called candidate solutions or feasible solutions.
The function is called, variously, an objective function, a loss function or cost function (minimization), a utility function or fitness function (maximization), or, in certain fields, an 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.
In mathematics, conventional optimization problems are usually stated in terms of minimization.
A local minimum is defined as an element for which there exists some such that
that is to say, on some region around 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, a global minimum is at least as good as every feasible element. Generally, unless the objective function is convex in a minimization problem, there may be several local minima. In a convex 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 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 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 nonconvex problem.
Optimization problems are often expressed with special notation. Here are some examples:
Consider the following notation:
Similarly, the notation
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:
This represents the value (or values) of the argument in the interval that minimizes (or minimize) the objective function (the actual minimum value of that function is not what the problem asks for). In this case, the answer is , since is infeasible, i.e. does not belong to the feasible set.
represents the pair (or pairs) that maximizes (or maximize) the value of the objective function , with the added constraint that lie in the interval (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form and , where ranges over all integers.
Operators and are sometimes also written as and , and stand for argument of the minimum and argument of the maximum.
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 John von Neumann developed the theory of duality in the same year.
Other notable researchers in mathematical optimization include the following:
In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):
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 cannot be improved upon according to one criterion without hurting another criterion 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. Evolutionary algorithms, however, are a very popular approach to obtain multiple solutions in a multi-modal optimization task.
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.
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.
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 satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.
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.
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 convex, then any local minimum will also be a global minimum. There exist efficient numerical techniques for minimizing convex functions, such as interior-point methods.
To solve problems, researchers may use algorithms that terminate in a finite number of steps, or iterative methods that converge to a solution (on some specified class of problems), or heuristics that may provide approximate solutions to some problems (although their iterates need not converge).
Optimization algorithms in machine learning
An optimization algorithm is a procedure which is executed iteratively by comparing various solutions until an optimum or a satisfactory solution is found. Optimization algorithms help us to minimize or maximize an objective function E(x) with respect to the internal parameters of a model mapping a set of predictors (X) to target values(Y). There are three types of optimization algorithms which are widely used; Zero order algorithms, First Order Optimization Algorithms and Second Order Optimization Algorithms.
Zero-order (or derivative-free) algorithms use only the criterion value at some positions. It is popular when the gradient and Hessian information are difficult to obtain, e.g., no explicit function forms are given.
First Order Optimization Algorithms
These algorithms minimize or maximize a Loss function E(x) using its Gradient values with respect to the parameters. Most widely used First order optimization algorithm is Gradient Descent. The First order derivative displays whether the function is decreasing or increasing at a particular point. First order Derivative basically will provide us a line which is tangential to a point on its Error Surface.
It is a first-order optimization algorithm for finding the minimum of a function.
θ=θ−η⋅∇J(θ) – this is the formula of the parameter updates, where ‘η’ is the learning rate, ’∇J(θ)’ is the Gradient of Loss function-J(θ) w.r.t parameters-‘θ’.
It is the most popular optimization algorithm used in optimizing a Neural Network. Gradient descent is used to update Weights in a Neural Network Model, i.e. update and tune the Model's parameters in a direction so that we can minimize the Loss function. A Neural Network trains via a technique called Back-propagation, in which propagating forward calculating the dot product of Inputs signals and their corresponding Weights and then applying an activation function to those sum of products, which transforms the input signal to an output signal and also is important to model complex Non-linear functions and introduces Non-linearity to the Model which enables the Model to learn almost any arbitrary functional mapping.
Second Order Optimization Algorithms
Second-order methods use the second order derivative which is also called Hessian to minimize or maximize the loss function. The Hessian is a Matrix of Second Order Partial Derivatives. Since the second derivative is costly to compute, the second order is not used much. The second order derivative informs us whether the first derivative is increasing or decreasing which hints at the function's curvature. It also provides us with a quadratic surface which touches the curvature of the Error Surface.
The iterative methods used to solve problems of nonlinear programming differ according to whether they evaluate Hessians, 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 the computational 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.
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 on line searches, which optimize a function along one dimension. A second and increasingly popular method for ensuring convergence uses trust regions. Both line searches and trust regions are used in modern methods of non-differentiable optimization. Usually a global optimizer is much slower than advanced local optimizers (such as BFGS), so often an efficient global optimizer can be constructed by starting the local optimizer from different starting points.
Besides (finitely terminating) algorithms and (convergent) iterative methods, there are heuristics. 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 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.
This approach may be applied in cosmology and astrophysics.
Economics is closely enough linked to optimization of agents that an influential definition relatedly describes economics qua science as the "study of human behavior as a relationship between ends and scarce means" with alternative uses. Modern optimization theory includes traditional optimization theory but also overlaps with game theory and the study of economic equilibria. The Journal of Economic Literature codes classify mathematical programming, optimization techniques, and related topics under JEL:C61-C63.
In microeconomics, the utility maximization problem and its dual problem, the expenditure minimization problem, are economic optimization problems. Insofar as they behave consistently, consumers are assumed to maximize their utility, while firms are usually assumed to maximize their profit. Also, agents are often modeled as being risk-averse, thereby preferring to avoid risk. Asset prices are also modeled using optimization theory, though the underlying mathematics relies on optimizing stochastic processes rather than on static optimization. International trade theory also uses optimization to explain trade patterns between nations. The optimization of portfolios is an example of multi-objective optimization in economics.
Since the 1970s, economists have modeled dynamic decisions over time using control theory. For example, dynamic search models are used to study labor-market behavior. A crucial distinction is between deterministic and stochastic models. Macroeconomists build dynamic 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.
Some common applications of optimization techniques in electrical engineering include active filter design, stray field reduction in superconducting magnetic energy storage systems, space mapping design of microwave structures, handset antennas, electromagnetics-based design. Electromagnetically validated design optimization of microwave components and antennas has made extensive use of an appropriate physics-based or empirical surrogate model and space mapping methodologies since the discovery of space mapping in 1993.
Optimization has been widely used in civil engineering. 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 and schedule optimization.
Another field that uses optimization techniques extensively is operations research. Operations research also uses stochastic modeling and simulation to support improved decision-making. Increasingly, operations research uses stochastic programming to model dynamic decisions that adapt to events; such problems can be solved with large-scale optimization and stochastic optimization methods.
Mathematical optimization is used in much modern controller design. High-level controllers such as model 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 in geophysical parameter estimation problems. Given a set of geophysical measurements, e.g. seismic recordings, it is common to solve for the physical properties and geometrical shapes of the underlying rocks and fluids.
Nonlinear optimization methods are widely used in conformational analysis.
Optimization techniques are used in many facets of computational systems biology such as model building, optimal experimental design, metabolic engineering, and synthetic biology. Linear programming has been applied to calculate the maximal possible yields of fermentation products, and to infer gene regulatory networks from multiple microarray datasets as well as transcriptional regulatory networks from high-throughput data. Nonlinear programming has been used to analyze energy metabolism and has been applied to metabolic engineering and parameter estimation in biochemical pathways.
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Methods to obtain suitable (in some sense) natural extensions of optimization problems that otherwise lack of existence or stability of solutions to obtain problems with guaranteed existence of solutions and their stability in some sense (typically under various perturbation of data) are in general called relaxation. Solutions of such extended (=relaxed) problems in some sense characterizes (at least certain features) of the original problems, e.g. as far as their optimizing sequences concerns. Relaxed problems may also possesses their own natural linear structure that may yield specific optimality conditions different from optimality conditions for the original problems.
Artelys Knitro is a commercial software package for solving large scale nonlinear mathematical optimization problems.
KNITRO – (the original solver name) short for "Nonlinear Interior point Trust Region Optimization" (the "K" is silent) – was co-created by Richard Waltz, Jorge Nocedal, Todd Plantenga and Richard Byrd. It was first introduced in 2001, as a derivative of academic research at Northwestern University (through Ziena Optimization LLC), and has since been continually improved by developers at Artelys.
Optimization problems must be presented to Knitro in mathematical form, and should provide a way of computing function derivatives using sparse matrices (Knitro can compute derivatives approximation but in most cases providing the exact derivatives is beneficial). An often easier approach is to develop the optimization problem in an algebraic modeling language. The modeling environment computes function derivatives, and Knitro is called as a "solver" from within the environment.BARON
BARON is a computational system for solving non-convex optimization problems to global optimality. Purely continuous, purely integer, and mixed-integer nonlinear problems can be solved with the software. BARON is available under the AIMMS, AMPL, and GAMS modeling languages on a variety of platforms. The GAMS/BARON solver is also available on the NEOS Server.The development of the BARON algorithms and software has been recognized by the 2004 INFORMS Computing Society Prize and the 2006 Beale-Orchard-Hays Prize for excellence in computational mathematical programming from the Mathematical Optimization Society.CPLEX
IBM ILOG CPLEX Optimization Studio (often informally referred to simply as CPLEX) is an optimization software package. In 2004, the work on CPLEX earned the first INFORMS Impact Prize.Cake number
In mathematics, the cake number, denoted by Cn, is the maximum number of regions into which a 3-dimensional cube can be partitioned by exactly n planes. The cake number is so-called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake.
The values of Cn for increasing n ≥ 0 are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, …(sequence A000125 in the OEIS)
The cake numbers are the 3-dimensional analogue of the 2-dimensional lazy caterer's sequence; the difference between successive cake numbers also gives the lazy caterer's sequence.Constraint (mathematics)
In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satisfy all constraints is called the feasible set.Convex optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. Whereas many classes of convex optimization problems admit polynomial-time algorithms, mathematical optimization is in general NP-hard. Convex optimization has applications in a wide range of disciplines, such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, finance, statistics (optimal experimental design), and structural optimization. With recent advancements in computing and optimization algorithms, convex programming is nearly as straightforward as linear programming.Discrete optimization
Discrete optimization is a branch of optimization in applied mathematics and computer science.Dual norm
In functional analysis, the dual norm is a measure of the "size" of each continuous linear functional defined on a normed vector space.GAUSS (software)
GAUSS is a matrix programming language for mathematics and statistics, developed and marketed by Aptech Systems. Its primary purpose is the solution of numerical problems in statistics, econometrics, time-series, optimization and 2D- and 3D-visualization. It was first published in 1984 for MS-DOS and is currently also available for Linux, macOS and Windows.Guess value
In mathematical modeling, a guess value is more commonly called a starting value or initial value. These are necessary for most optimization problems which use search algorithms, because those algorithms are mainly deterministic and iterative, and they need to start somewhere. One common type of application is nonlinear regression.Gurobi
The Gurobi Optimizer is a commercial optimization solver for linear programming (LP), quadratic programming (QP), quadratically constrained programming (QCP), mixed integer linear programming (MILP), mixed-integer quadratic programming (MIQP), and mixed-integer quadratically constrained programming (MIQCP).
Gurobi is named for its founders: Zonghao Gu, Edward Rothberg and Robert Bixby. Bixby was also the founder of CPLEX, while Rothberg and Gu led the CPLEX development team for nearly a decade.Lazy caterer's sequence
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a disk (a pancake or pizza is usually used to describe the situation) that can be made with a given number of straight cuts. For example, three cuts across a pancake will produce six pieces if the cuts all meet at a common point inside the circle, but up to seven if they do not. This problem can be formalized mathematically as one of counting the cells in an arrangement of lines; for generalizations to higher dimensions, see arrangement of hyperplanes.
The analogue of this sequence in three dimensions is the cake number.List of optimization software
Given a transformation between input and output values, described by a mathematical function f, optimization deals with generating and selecting a best solution from some set of available alternatives, by systematically choosing input values from within an allowed set, computing the output of the function, and recording the best output values found during the process. Many real-world problems can be modeled in this way. For example, the inputs can be design parameters of a motor, the output can be the power consumption, or the inputs can be business choices and the output can be the obtained profit.
An optimization problem, in this case a minimization problem, can be represented in the following way
In continuous optimization, 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. In combinatorial optimization, A is some subset of a discrete space, like binary strings, permutations, or sets of integers.
The use of optimization software requires that the function f is defined in a suitable programming language and connected at compile or run time to the optimization software. The optimization software will deliver input values in A, the software module realizing f will deliver the computed value f(x) and, in some cases, additional information about the function like derivatives.
In this manner, a clear separation of concerns is obtained: different optimization software modules can be easily tested on the same function f, or a given optimization software can be used for different functions f.
The following tables provide a list of notable optimization software organized according to license and business model type.MINOS (optimization software)
MINOS is a Fortran software package for solving linear and nonlinear mathematical optimization problems. MINOS (Modular In-core Nonlinear Optimization System) may be used for linear programming, quadratic programming, and more general objective functions and constraints, and for finding a feasible point for a set of linear or nonlinear equalities and inequalities.MINOS was first developed by Bruce Murtagh and Michael Saunders, mostly at the Systems Optimization Laboratory in the Department of Operations Research at Stanford University. In 1985, Saunders was awarded the inaugural Orchard-Hays prize by the Mathematical Programming Society (now the Mathematical Optimization Society) for his work on MINOS. Despite being one of the first general-purpose constrained optimization solvers to emerge, the package remains heavily used. MINOS is supported in the AIMMS, AMPL, APMonitor, GAMS, and TOMLAB modeling systems. In addition, it remains one of the top-used solvers on the NEOS Server and in GAMS.MOSEK
MOSEK is a software package for the solution of linear, mixed-integer linear, quadratic, mixed-integer quadratic, quadratically constraint, conic and convex nonlinear mathematical optimization problems. The emphasis in MOSEK is on solving large scale sparse problems. Particularly the interior-point optimizer for linear, conic quadratic (a.k.a. Second-order cone programming) and semi-definite (aka. semidefinite programming) problems is very efficient. A special feature of the MOSEK interior-point optimizer is that it is based on the so-called homogeneous model which implies MOSEK can reliably detect a primal and/or dual infeasible status as documented in several published papers.The software is developed by Mosek ApS, a danish company with its office in Copenhagen, the capital of Denmark.
In addition to the interior-point optimizer MOSEK includes:
Primal and dual simplex optimizer for linear problems.
A primal network simplex optimizer for problems with special network structure.
Mixed-integer optimizer for linear, quadratic and conic quadratic problems.MOSEK provides interfaces to the C, C#, Java and Python languages. Most major modeling systems are made compatible for MOSEK, examples are: AMPL, and GAMS.
MOSEK can also be used from popular tools such as MATLAB, R (an outdated version of package Rmosek is available from the CRAN server, the up-to-date version is provided by Mosek ApS), CVX, and YALMIP.Mathematical Optimization Society
The Mathematical Optimization Society (MOS), known as the Mathematical Programming Society until 2010, is an international association of researchers active in optimization. The MOS encourages the research, development, and use of optimization—including mathematical theory, software implementation, and practical applications (operations research).
Founded in 1973, the MOS has several activities: Publishing journals and a newsletter, organizing and cosponsoring conferences, and awarding prizes.Maxima and minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.Process optimization
Process optimization is the discipline of adjusting a process so as to optimize (make the best or most effective use of) some specified set of parameters without violating some constraint. The most common goals are minimizing cost and maximizing throughput and/or efficiency. This is one of the major quantitative tools in industrial decision making.
When optimizing a process, the goal is to maximize one or more of the process specifications, while keeping all others within their constraints. This can be done by using a process mining tool, discovering the critical activities and bottlenecks, and acting only on them.TOMLAB
The TOMLAB Optimization Environment is a modeling platform for solving applied optimization problems in MATLAB.