In mathematics, computer science and operations research, mathematical
optimization or mathematical programming, alternatively spelled
optimisation, 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
1 Optimization problems
2.1 Minimum and maximum value of a function
2.2 Optimal input arguments
4 Major subfields
4.1 Multi-objective optimization
4.2 Multi-modal optimization
5 Classification of critical points and extrema
5.1 Feasibility problem
5.3 Necessary conditions for optimality
5.4 Sufficient conditions for optimality
5.5 Sensitivity and continuity of optima
Calculus of optimization
6 Computational optimization techniques
6.1 Optimization algorithms
6.2 Iterative methods
6.3 Global convergence
Economics and finance
7.3 Electrical engineering
7.4 Civil engineering
7.5 Operations research
7.6 Control engineering
7.8 Molecular modeling
9 See also
11 Further reading
11.1.1 Undergraduate level
11.1.2 Graduate level
11.2 Continuous optimization
11.3 Combinatorial optimization
11.4 Relaxation (extension method)
13 External links
Main article: Optimization problem
An optimization problem can be represented in the following way:
Given: a function f : A
R from some set A to the real numbers
Sought: an element x0 in A such that f(x0) ≤ f(x) for all x in A
("minimization") or such that f(x0) ≥ f(x) for all x in A
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 f as representing the energy of the system being modeled.
Typically, A is some subset of the
Euclidean space Rn, often specified
by a set of constraints, equalities or inequalities that the members
of A have to satisfy. The domain A of f is called the search space or
the choice set, while the elements of A are called candidate solutions
or feasible solutions.
The function f is called, variously, an objective function, 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
In mathematics, conventional optimization problems are usually stated
in terms of minimization. Generally, unless both the objective
function and the feasible region are convex in a minimization problem,
there may be several local minima. A local minimum x* is defined as a
point for which there exists some δ > 0 such that for all x where
displaystyle mathbf x -mathbf x ^ * leq delta ,,
displaystyle f(mathbf x ^ * )leq f(mathbf x )
holds; that is to say, on some region around x* all of the function
values are greater than or equal to the value at that point. Local
maxima are defined similarly.
While a local minimum is at least as good as any nearby points, a
global minimum is at least as good as every feasible point. In a
convex problem, if there is a local minimum that is interior (not on
the edge of the set of feasible points), 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:
Minimum and maximum value of a function
Consider the following notation:
displaystyle min _ xin mathbb R ;(x^ 2 +1)
This denotes the minimum value of the objective function
displaystyle x^ 2 +1
, when choosing x from the set of real numbers
displaystyle mathbb R
. The minimum value in this case is
, occurring at
Similarly, the notation
displaystyle max _ xin mathbb R ;2x
asks for the maximum value of the objective function 2x, where x may
be any real number. In this case, there is no such maximum as the
objective function is unbounded, so the answer is "infinity" or
Optimal input arguments
Main article: Arg max
Consider the following notation:
displaystyle underset xin (-infty ,-1] operatorname arg,min
;x^ 2 +1,
displaystyle underset x operatorname arg,min ;x^ 2 +1,;
text subject to: ;xin (-infty ,-1].
This represents the value (or values) of the argument x in the
displaystyle (-infty ,-1]
that minimizes (or minimize) the objective function x2 + 1
(the actual minimum value of that function is not what the problem
asks for). In this case, the answer is x = –1, since x = 0 is
infeasible, i.e. does not belong to the feasible set.
displaystyle underset xin [-5,5],;yin mathbb R operatorname
displaystyle underset x,;y operatorname arg,max ;xcos(y),;
text subject to: ;xin [-5,5],;yin mathbb R ,
pair (or pairs) that maximizes (or maximize) the value of the
, with the added constraint that x 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 (5, 2kπ) and
(−5,(2k+1)π), where k ranges over all integers.
arg min and arg max are sometimes also written argmin and argmax, and
stand for argument of the minimum and argument of the maximum.
Fermat and Lagrange found calculus-based formulae for identifying
optima, while Newton and Gauss proposed iterative methods for moving
towards an optimum.
The term "linear programming" for certain optimization cases was due
to George B. Dantzig, although much of the theory had been
Leonid Kantorovich in 1939. (Programming in this context
does not refer to computer programming, but 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 major researchers in mathematical optimization include the
Ronald A. Howard
R. Tyrrell Rockafellar
Naum Z. Shor
Michael J. Todd
Convex programming studies the case when the objective function is
convex (minimization) or concave (maximization) and the constraint set
is convex. This can be viewed as a particular case of nonlinear
programming or as generalization of linear or convex quadratic
Linear programming (LP), a type of convex programming, studies the
case in which the objective function f is linear and the constraints
are specified using only linear equalities and inequalities. Such a
constraint set is called a polyhedron or a polytope if it is bounded.
Second order cone programming (SOCP) is a convex program, and includes
certain types of quadratic programs.
Semidefinite programming (SDP) is a subfield of convex optimization
where the underlying variables are semidefinite matrices. It is a
generalization of linear and convex quadratic programming.
Conic programming is a general form of convex programming. LP, SOCP
and SDP can all be viewed as conic programs with the appropriate type
Geometric programming is a technique whereby objective and inequality
constraints expressed as posynomials and equality constraints as
monomials can be transformed into a convex program.
Integer programming studies linear programs in which some or all
variables are constrained to take on integer values. This is not
convex, and in general much more difficult than regular linear
Quadratic programming allows the objective function to have quadratic
terms, while the feasible set must be specified with linear equalities
and inequalities. For specific forms of the quadratic term, this is a
type of convex programming.
Fractional programming studies optimization of ratios of two nonlinear
functions. The special class of concave fractional programs can be
transformed to a convex optimization problem.
Nonlinear programming studies the general case in which the objective
function or the constraints or both contain nonlinear parts. This may
or may not be a convex program. In general, whether the program is
convex affects the difficulty of solving it.
Stochastic programming studies the case in which some of the
constraints or parameters depend on random variables.
Robust programming is, like stochastic programming, an attempt to
capture uncertainty in the data underlying the optimization problem.
Robust optimization targets to find solutions that are valid under all
possible realizations of the uncertainties.
Combinatorial optimization is concerned with problems where the set of
feasible solutions is discrete or can be reduced to a discrete one.
Stochastic optimization is used with random (noisy) function
measurements or random inputs in the search process.
Infinite-dimensional optimization studies the case when the set of
feasible solutions is a subset of an infinite-dimensional space, such
as a space of functions.
Heuristics and metaheuristics make few or no assumptions about the
problem being optimized. Usually, heuristics do not guarantee that any
optimal solution need be found. On the other hand, heuristics are used
to find approximate solutions for many complicated optimization
Constraint satisfaction studies the case in which the objective
function f is constant (this is used in artificial intelligence,
particularly in automated reasoning).
Constraint programming is a programming paradigm wherein relations
between variables are stated in the form of constraints.
Disjunctive programming is used where at least one constraint must be
satisfied but not all. It is of particular use in scheduling.
Space mapping is a concept for modeling and optimization of an
engineering system to high-fidelity (fine) model accuracy exploiting a
suitable physically meaningful coarse or surrogate model.
In a number of subfields, the techniques are designed primarily for
optimization in dynamic contexts (that is, decision making over time):
Calculus of variations seeks to optimize an action integral over some
space to an extremum by varying a function of the coordinates.
Optimal control theory is a generalization of the calculus of
variations which introduces control policies.
Dynamic programming 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 the Bellman equation.
Mathematical programming with equilibrium constraints is where the
constraints include variational inequalities or complementarities.
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 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
Classification of critical points and extrema
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
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.
Necessary conditions for optimality
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
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
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 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.
Sensitivity and continuity of optima
The envelope theorem describes how the value of an optimal solution
changes when an underlying parameter changes. The process of computing
this change is called comparative statics.
The maximum theorem of
Claude Berge (1963) describes the continuity of
an optimal solution as a function of underlying parameters.
Calculus of optimization
Main article: Karush–Kuhn–Tucker conditions
See also: Critical point (mathematics), Differential calculus,
Gradient, Hessian matrix, Positive definite matrix, Lipschitz
continuity, Rademacher's theorem, Convex function, and Convex analysis
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
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.
Computational optimization techniques
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).
See also: List of optimization algorithms
Simplex algorithm of George Dantzig, designed for linear programming.
Extensions of the simplex algorithm, designed for quadratic
programming and for linear-fractional programming.
Variants of the simplex algorithm that are especially suited for
Quantum optimization algorithms
Main article: Iterative method
See also: Newton's method in optimization, Quasi-Newton method, Finite
difference, Approximation theory, and Numerical analysis
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
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
derivates, 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, using finite
Sequential quadratic programming: A Newton-based method for
small-medium scale constrained problems. Some versions can handle
Interior point methods: This is a large class of methods for
constrained optimization. Some interior-point methods use only
(sub)gradient information, and others of which require the evaluation
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
Subgradient methods - An iterative method for large locally Lipschitz
functions using generalized gradients. Following Boris T. Polyak,
subgradient–projection methods are similar to conjugate–gradient
Bundle method of descent: An iterative method for small–medium-sized
problems with locally Lipschitz functions, particularly for convex
minimization problems. (Similar to conjugate gradient methods)
Ellipsoid method: An iterative method for small problems with
quasiconvex objective functions and of great theoretical interest,
particularly in establishing the polynomial time complexity of some
combinatorial optimization problems. It has similarities with
Reduced gradient method (Frank–Wolfe) for approximate minimization
of specially structured problems with linear 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).
Iterative methods for medium-large problems
Simultaneous perturbation stochastic approximation (SPSA) method for
stochastic optimization; uses random (efficient) gradient
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.
Pattern search methods, which have better convergence properties than
the Nelder–Mead heuristic (with simplices), which is listed below.
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.
Main article: Heuristic algorithm
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
Hill climbing with random restart
Nelder-Mead simplicial heuristic: A popular heuristic for approximate
minimization (without calling gradients)
Particle swarm optimization
Gravitational search algorithm
Artificial bee colony optimization
Reactive Search Optimization (RSO) implemented in LIONsolver
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
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
This approach may be applied in cosmology and astrophysics,.
Economics and finance
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
International trade theory
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, microeconomists use dynamic search
models 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
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
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 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.
Main article: Molecular modeling
Nonlinear optimization methods are widely used in conformational
Main article: List of optimization software
Deterministic global optimization
Important publications in optimization
Mathematical Optimization Society (formerly Mathematical Programming
Mathematical optimization algorithms
Mathematical optimization software
Test functions for optimization
Vehicle routing problem
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optimization (pp. 1–72);
Donald Goldfarb and Michael J. Todd, Linear programming
Philip E. Gill, Walter Murray, Michael A. Saunders, and Margaret H.
Wright, Constrained nonlinear programming (pp. 171–210);
Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin, Network
flows (pp. 211–369);
W. R. Pulleyblank, Polyhedral combinatorics (pp. 371–446);
George L. Nemhauser
George L. Nemhauser and Laurence A. Wolsey,
Claude Lemaréchal, Nondifferentiable optimization
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A. H. G. Rinnooy Kan and G. T. Timmer, Global optimization
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Xin-She Yang (2010). Engineering Optimization: An Introduction with
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Relaxation (extension method)
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
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Cambridge Univ. Press, 1999.
P. Pedregal: Parametrized Measures and Variational Principles.
Birkhäuser, Basel, 1997
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Calculus". W. de Gruyter, Berlin, 1997. ISBN 3-11-014542-1.
Optimal control of differential and functional equations.
Academic Press, 1972.
Computational Optimization and Applications
Journal of Computational Optimization in
Economics and Finance
Journal of Economic Dynamics and Control
SIAM Journal on Optimization (SIOPT) and Editorial Policy
SIAM Journal on Control and Optimization (SICON) and Editorial Policy
COIN-OR—Computational Infrastructure for Operations Research
Decision Tree for Optimization Software Links to optimization source
Mathematical Programming Glossary
Mathematical Programming Society
NEOS Guide currently being replaced by the NEOS Wiki[dead link]
Optimization Online A repository for optimization e-prints
Optimization Related Links
Convex Optimization I EE364a: Course from Stanford University
Convex Optimization – Boyd and Vandenberghe Book on Convex
Book and Course on Optimization Methods for Engineering Design
Mathematical Optimization in Operations Research from the Institute
for Operations Research and the Management Sciences (INFORMS)
Optimization: Algorithms, methods, and heuristics
Unconstrained nonlinear: Methods calling …
Successive parabolic interpolation
… and gradients
BFGS and L-BFGS
Symmetric rank-one (SR1)
… and Hessians
Augmented Lagrangian methods
Sequential quadratic programming
Successive linear programming
Reduced gradient (Frank–Wolfe)
Ellipsoid algorithm of Khachiyan
Projective algorithm of Karmarkar
Simplex algorithm of Dantzig
Revised simplex algorithm
Principal pivoting algorithm of Lemke
Branch and bound/cut
Push–relabel maximum flow
Areas of mathematics
History of mathematics
Philosophy of mathematics
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Differential equations / Dynamical systems
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Verification and validation
Systems development life cycle
Quality function deployment
Systems Modeling Language
Work breakdown structure
James S. Albus
Benjamin S. Blanchard
Wernher von Braun
Arthur David Hall III
Robert E. Machol
Joseph Francis Shea
Manuela M. Veloso
John N. Warfield