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Mathematical optimization concept
The dual of a given linear program (LP) is another LP that is derived from the original (the primal) LP in the following schematic way: Each variable in
Dual_linear_program
Method to solve optimization problems
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical
Linear_programming
Principle in mathematical optimization
formal by the equations in Linear programming: Duality. In nonlinear programming, the constraints are not necessarily linear. Nonetheless, many of the
Duality_(optimization)
Concept from linear programming
In the theory of linear programming, a basic feasible solution (BFS) is a solution with a minimal set of non-zero variables. Geometrically, each BFS corresponds
Basic_feasible_solution
Subfield of convex optimization
Semidefinite programming (SDP) is a subfield of mathematical programming concerned with the optimization of a linear objective function (a user-specified
Semidefinite_programming
Condition in mathematical optimization
implies polynomial-time solvability. Convex optimization Linear programming#Duality Dual linear program Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis
Strong_duality
Concept in mathematical optimization
linear-fractional programming (LFP) is a generalization of linear programming (LP). Whereas the objective function in a linear program is a linear function
Linear-fractional_programming
General concept and operation in mathematics
duality Langlands dual Linear programming#Duality List of dualities Matlis duality Petrie duality Pontryagin duality S-duality T-duality, Mirror symmetry
Duality_(mathematics)
Subset of a graph's vertices, including at least one endpoint of every edge
vertex cover problem can be formulated as a half-integral, linear program whose dual linear program is the maximum matching problem. Vertex cover problems
Vertex_cover
Equivalence of optimization problems
flow of water to the city. This is a special case of the duality theorem for linear programs and can be used to derive Menger's theorem and the Kőnig–Egerváry
Max-flow_min-cut_theorem
Solving an optimization problem with a quadratic objective function
function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming. "Programming" in this context refers
Quadratic_programming
On bipartite matching and vertex cover
cover LP is exactly the dual linear program of the maximum fractional matching LP. Therefore, by the LP duality theorem, both programs have the same solution
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_theory)
Software package
The GNU Linear Programming Kit (GLPK) is a software package intended for solving large-scale linear programming (LP), mixed integer programming (MIP),
GNU_Linear_Programming_Kit
System of resource-aware logic
particularly because of its emphasis on resource-boundedness, duality, and interaction. Linear logic lends itself to many different presentations, explanations
Linear_logic
Supervised boosting classification model
primal linear program corresponds to rows in the dual linear program. The equivalent dual linear program of LPBoost is the following linear program. max
LPBoost
Linear programming for Combinatorial optimization
The configuration linear program (configuration-LP) is a linear programming technique used for solving combinatorial optimization problems. It was introduced
Configuration_linear_program
Gives conditions that guarantee the max–min inequality holds with equality
Parthasarathy's theorem – a generalization of Von Neumann's minimax theorem Dual linear program can be used to prove the minimax theorem for zero-sum games. Yao's
Minimax_theorem
Concept in optimization
certain cases. Many primal-dual approximation algorithms are based on the principle of weak duality. Consider a linear programming problem, where A {\displaystyle
Weak_duality
totally dual integral (TDI) if for any c ∈ Z n {\displaystyle c\in \mathbb {Z} ^{n}} such that there is a feasible, bounded solution to the linear program max
Total_dual_integrality
Solvability theorem for finite systems of linear inequalities
linear programming duality and has played a central role in the development of mathematical optimization (alternatively, mathematical programming).[citation
Farkas'_lemma
Mathematical model for sequential decision making under uncertainty
i,a)h(j)\geq R(i,a)\,\,\forall i\in S,\,a\in A(i)\end{aligned}}} Dual linear program(D-LP) Maximize ∑ i ∈ S ∑ a ∈ A ( i ) R ( i , a ) y ( i , a ) s.t
Markov_decision_process
Mathematical convex optimization
prototypical primal and dual semidefinite program is a minimization of a real linear function respectively subject to the primal and dual convex cones governing
Linear_matrix_inequality
Algorithm for solving linear programs
an efficient algorithm for solving large linear programs. The overarching idea is that many linear programs are too large to consider all the variables
Column_generation
Subfield of mathematical optimization
function f is a linear function. This is because any program with a general objective can be transformed into a program with a linear objective by adding
Convex_optimization
Context dependence in quantum measurements
the algebraic maximum violation of the inequality. Moreover, the dual linear program to that which maximises λ computes a noncontextual inequality for
Quantum_contextuality
Structure in convex geometry
P is a polyhedral fan that is dual to P. Normal fans have applications to polyhedral combinatorics, linear programming, tropical geometry, toric geometry
Normal_fan
Classical problem in combinatorics
primal-dual algorithm for the set cover problem is an iterative method that constructs feasible solutions to both the primal and dual linear programs simultaneously
Set_cover_problem
Subfield of convex optimization
the dual cone of C {\displaystyle C\ } . Whilst weak duality holds in conic linear programming, strong duality does not necessarily hold. The dual of
Conic_optimization
Equivalence of average-case and expected complexity
minimax theorem in the theory of zero-sum games, and to the duality theory of linear programs. Yao's principle is formulated in terms of an arbitrary real
Yao's_principle
Algorithms for solving convex optimization problems
the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programming called Karmarkar's algorithm, which runs in polynomial time ( O (
Interior-point_method
Algorithm for solving linear programming problems with special structure
Dantzig–Wolfe decomposition is an algorithm for solving (mixed integer) linear programming problems by exploiting their structure. It was originally developed
Dantzig–Wolfe_decomposition
Branch of mathematics
Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b
Linear_algebra
Algorithm in graph theory
200 to 300 times faster than the simplex method applied to general linear program of same dimensions. For a long time, the existence of a provably efficient
Network_simplex_algorithm
optimization, fractional programming is a generalization of linear-fractional programming. The objective function in a fractional program is a ratio of two functions
Fractional_programming
Convex optimization problem
equivalent to a convex quadratically constrained linear program. Convex quadratically constrained quadratic programs can also be formulated as SOCPs by reformulating
Second-order_cone_programming
Iterative method for minimizing convex functions
this is by combining the primal and dual linear programs together into one program, and adding the additional (linear) constraint that the value of the
Ellipsoid_method
Set of related approximation algorithms for the bin packing problem
number of variables - a variable for each possible configuration. The dual linear program of the fractional LP is: maximize n ⋅ y s.t. A T y
Karmarkar–Karp bin packing algorithms
Karmarkar–Karp_bin_packing_algorithms
Optimization algorithm
}} are the primal and dual displacements, respectively. Note that the Lagrangian Hessian is not explicitly inverted and a linear system is solved instead
Sequential quadratic programming
Sequential_quadratic_programming
Optimization problem in mathematics
,Pm are all zero, then the constraints are in fact linear and the problem is a quadratic program. A convex QCQP problem can be efficiently solved using
Quadratically constrained quadratic program
Quadratically_constrained_quadratic_program
Optimization technique for solving (mixed) integer linear programs
means of linear inequalities, termed cuts. Such procedures are commonly used to find integer solutions to mixed integer linear programming (MILP) problems
Cutting-plane_method
American mathematician (1924–2021)
including a delightful abstraction of linear programming duality (1963). He also continued to use properties of linear inequalities to prove (or re-prove
Alan_J._Hoffman
Mathematical concept
variable is also added. Slack variables are used in particular in linear programming. As with the other variables in the augmented constraints, the slack
Slack_variable
Optimization software package
sparse problems linear and conic optimization problems. In particular, MOSEK solves conic quadratic (a.k.a. Second-order cone programming) and semi-definite
MOSEK
Branch of mathematics that studies abstract algebraic structures
studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an
Representation_theory
Filename extension for a native executable program
it is possible to specify a custom DOS stub. Indeed, there are a few dual programs, such as regedit in Windows 95 and old versions of WinZIP self extractors
.exe
LINDO – (Linear, Interactive, and Discrete optimizer) a software package for linear programming, integer programming, nonlinear programming, stochastic
List_of_optimization_software
Mathematical set closed under positive linear combinations
In linear algebra, a cone—sometimes called a linear cone to distinguish it from other sorts of cones—is a subset of a real vector space that is closed
Convex_cone
Type of voltage regulator
In electronics, a linear regulator is a voltage regulator used to maintain a steady voltage. The resistance of the regulator varies in accordance with
Linear_regulator
Suite of mathematical modeling and optimization tools
commercial optimization solver for linear programming (LP), mixed integer linear programming (MILP), convex quadratic programming (QP), convex quadratically constrained
FICO_Xpress
Method for mathematical optimization
family of algorithms for linear programming. Variants of the criss-cross algorithm also solve more general problems with linear inequality constraints and
Criss-cross_algorithm
Benson, is a method for solving multi-objective linear programming problems and vector linear programs. This works by finding the "efficient extreme points
Benson's_algorithm
Graph coloring where graph elements are assigned sets of colors
is at most 1. The strong duality theorem of linear programming guarantees that the optimal solutions to both linear programs have the same value. Note
Fractional_coloring
Google program
GLOP (the Google Linear Optimization Package) is Google's open-source linear programming solver, created by Google's Operations Research Team. It is written
GLOP
American mathematician (1914–2005)
algorithm, an algorithm for solving linear programming problems, and for his other work with linear programming. In statistics, Dantzig solved two open
George_Dantzig
Type of computational problem
Covering problems are minimization problems and usually integer linear programs, whose dual problems are called packing problems. The most prominent examples
Covering_problems
Primal-Dual algorithm optimization for convex problems
algorithm in PyTorch for GPU-accelerated linear programming in his Primal-Dual Algorithm for Linear Programming GitHub Repository The Manopt.jl package
Chambolle–Pock_algorithm
Numerical software
open-source software to solve linear programming (LP), mixed-integer programming (MIP), and convex quadratic programming (QP) models. Written in C++ and
HiGHS_optimization_solver
Research program on the symmetries of geometry
by linear fractional transformations and a large portion of the respective geometries can be obtained in a uniform way from the Erlangen program. Some
Erlangen_program
Concepts in convex analysis
Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics. The dual cone C* of a subset C in a linear space X
Dual_cone_and_polar_cone
Algorithm for solving linear programming problems
mathematical optimization, affine scaling is an algorithm for solving linear programming problems. Specifically, it is an interior point method, discovered
Affine_scaling
Type of rocket engine
shorter truncated spike (like base bleed in an artillery shell). In the linear aerospike the spike consists of a tapered wedge-shaped plate, with exhaust
Aerospike_engine
Economical computational problem
establishes that the game value can be computed by solving dual linear programs. Since linear programming can be solved in polynomial time using interior-point
Nash_equilibrium_computation
Class of error-correcting code
In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally
Linear_code
Conjectures connecting number theory and geometry
general linear group GL(n), other connected reductive groups can be used. Furthermore, given such a group G, Langlands constructs the Langlands dual group
Langlands_program
Non-destructive audio, video, or image editing
as the audio, video, or image is edited. A non-linear editing system is a video editing (NLVE) program or application, or an audio editing (NLAE) digital
Non-linear_editing
Abstraction of ordered linear algebra
Rockafellar 1984 and 1998. Bachem, Achim; Kern, Walter (1992). Linear Programming Duality: An Introduction to Oriented Matroids. Universitext. Springer-Verlag
Oriented_matroid
Optimization software package for linear programming
CPLEX Optimizer solves integer programming problems, very large linear programming problems using either primal or dual variants of the simplex method
CPLEX
Fleet of truck-mounted weather radars
4 was completed in 2000, becoming the first Doppler on Wheels to feature dual polarization, allowing it to scan using horizontal and vertical polarizations
Doppler_on_Wheels
Type of engine with no crank
cycle-to-cycle variations were reported for dual piston engines. In June 2014 Toyota announced a prototype Free Piston Engine Linear Generator (FPEG). As the piston
Free-piston_engine
Digital optical disc data storage format
standard single-layer DVD can store up to 4.7 GB of data, a dual-layer DVD up to 8.5 GB. Dual-layer, double-sided DVDs can store up to a maximum of 17.08
DVD
Abstraction of linear independence of vectors
/ˈmeɪtrɔɪd/ is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define
Matroid
Class of algorithms for solving constrained optimization problems
[citation needed] Sequential quadratic programming Sequential linear programming Sequential linear-quadratic programming Open source and non-free/commercial
Augmented_Lagrangian_method
Theorem in graph theory
version, and which in turn is a special case of the strong duality theorem for linear programs. The edge-connectivity version of Menger's theorem is as
Menger's_theorem
Set of functions between two fixed sets
the set of linear maps X → V form a vector space over F with pointwise operations (often denoted Hom(X,V)). One such space is the dual space of X: the
Function_space
Robotics problem on coordinating two parts of a robot
formulating the matrices as dual quaternions, it is possible to get a linear equation by which X is solvable in a linear format. An alternative way applies
Hand–eye_calibration_problem
Annual entrance test held in India
5-year BS-MS Dual Degree Programs of the IISERs 5-year BS-MS in Computational and Data Sciences of IISER Kolkata (from 2025) 4-year B.Tech Programs (Chemical
IISER_Aptitude_Test
Set of methods for supervised statistical learning
is called the dual problem. Since the dual maximization problem is a quadratic function of the c i {\displaystyle c_{i}} subject to linear constraints,
Support_vector_machine
Magnetic tape data storage technology
renamed Digital Linear Tape (DLT) and eventually sold to Quantum Corporation. In the late 1980s, Exabyte's Data8 format, derived from Sony's dual-reel cartridge
Linear_Tape-Open
Modal temporal logic with modalities referring to time
In logic, linear temporal logic or linear-time temporal logic (LTL) is a modal temporal logic with modalities referring to time. In LTL, one can encode
Linear_temporal_logic
Lagrangian duality Fenchel duality Philip Wolfe (1961). "A duality theorem for non-linear programming". Quarterly of Applied Mathematics. 19 (3): 239–244. doi:10
Wolfe_duality
mathematics, the duality gap is the difference between the primal and dual solutions. If d ∗ {\displaystyle d^{*}} is the optimal dual value and p ∗ {\displaystyle
Duality_gap
Problem in formal logic
Jean H. (1984), "Linear-time algorithms for testing the satisfiability of propositional Horn formulae", Journal of Logic Programming, 1 (3): 267–284,
Horn-satisfiability
Method in mathematical optimization
function of the dual variables (the Lagrangian multipliers) is the Lagrangian dual problem. Suppose we are given a linear programming problem, with x
Lagrangian_relaxation
called a generalized linear program) is an optimization problem that shares certain properties with low-dimensional linear programs and that may be solved
LP-type_problem
optimization problem Total dual integrality — concept of duality for integer linear programming Wolfe duality — for when objective function and constraints are
List of numerical analysis topics
List_of_numerical_analysis_topics
Core Functionality". Retrieved 2026-06-16. OR/MS Today: 2013 Linear Programming Software Survey OR/MS Today: 1998 Nonlinear Programming Software Survey
Comparison of optimization software
Comparison_of_optimization_software
Equivalence of two physical theories
Montonen–Olive duality is closely related to a research program in mathematics called the geometric Langlands program. Another realization of S-duality in quantum
S-duality
Algebraic object with geometric applications
is always a natural linear map from V to its double dual, given by evaluating a linear form in V∗ against a vector in V. This linear mapping is an isomorphism
Tensor
Linear production game (LP Game) is a N-person game in which the value of a coalition can be obtained by solving a linear programming problem. It is widely
Linear_production_game
Optimization algorithm
Krister Svanberg in the 1980s. It's primarily used for solving non-linear programming problems, particularly those related to structural design and topology
Method_of_moving_asymptotes
Technique in mathematical optimization
large linear programming problems that have a special block structure. This block structure often occurs in applications such as stochastic programming as
Benders_decomposition
Situation where total gains match total losses
often solved with the minimax theorem which is closely related to linear programming duality, or with Nash equilibrium. In contrast, positive-sum or win–win
Zero-sum_game
Optimization algorithm
feasible set is given by a set of linear constraints, then the subproblem to be solved in each iteration becomes a linear program. While the worst-case convergence
Frank–Wolfe_algorithm
Strong-weak duality in supersymmetric theories of theoretical physics
Montonen–Olive duality or electric–magnetic duality is the oldest known example of strong–weak duality or S-duality according to current terminology. It
Montonen–Olive_duality
Mathematical operation on vector spaces
s}.} Here V ∗ {\displaystyle V^{*}} is the dual vector space (which consists of all linear maps f from V to the ground field K). There is a product
Tensor_product
American researcher and mathematician
Lizhen (April 2012). "A dual variant of Benson's 'outer approximation algorithm' for multiple objective linear programming". Journal of Global Optimization
Harold_Benson
Class of algorithms that find approximate solutions to optimization problems
appropriate mathematical programming formulation (typically a convex programming) such as Linear programming, Semidefinite programming, etc, to obtain a relaxation
Approximation_algorithm
Class of supercarriers for the U.S. Navy
AN/SPY-4 S Band volume search radar on the lead ship. Designated together as Dual Band Radar (DBR), initially developed for the Zumwalt-class destroyers. Starting
Gerald R. Ford-class aircraft carrier
Gerald_R._Ford-class_aircraft_carrier
Möbius transformation generalized to rings other than the complex numbers
In mathematics, a linear fractional transformation is, roughly speaking, an invertible transformation of the form z ↦ a z + b c z + d . {\displaystyle
Linear fractional transformation
Linear_fractional_transformation
Device that compares two voltages or currents
Voltage Dual Comparators". Linear Technology. Archived from the original on 2011-01-02. "MAX9021, MAX9022, MAX9024: Micropower, Ultra-Small, Single/Dual/Quad
Comparator
DUAL LINEAR-PROGRAM
DUAL LINEAR-PROGRAM
Boy/Male
Hindu
Lingam
Surname or Lastname
English
English : metronymic from Line.
Girl/Female
Tamil
Dual, Second
Female
English
Variant spelling of English Linsey, LINSAY means "Lincoln's wetlands."
Male
English
Variant spelling of English Dougal, DUGAL means "black stranger."Â
Boy/Male
Hindu
Dear one
Girl/Female
Muslim
Prayer
Surname or Lastname
English
English : habitational name from Lingart, Lancashire, or Lingards Wood in Marsden, West Yorkshire, both named from Old English līn ‘flax’ + garðr ‘enclosure’.
Male
Yiddish
 Variant spelling of Yiddish Lieber, LIBER means "beloved." Compare with another form of Liber.
Boy/Male
French
Of the valley.
Male
Scandinavian
Scandinavian form of Old Norse Einarr, EINAR means "lone warrior."
Surname or Lastname
English
English : variant of Dale (from the Old Kentish form del) or a habitational name from Deal in Kent, named with this word.Americanized spelling of German Diel or Diehl.Dutch (de Ruyter) : variant spelling (17th century) of De Ruiter
Surname or Lastname
French
French : topographic name from Old French du val ‘from the valley’ (from Latin vallis).English : variant of Duvall 1.
Surname or Lastname
English
English : of uncertain origin; possibly an altered form of Irish Doyle. Compare Dial.Indian : variant spelling of Dayal.
Male
English
Irish Anglicized form of Gaelic Fionnbarr, FINBAR means "fair-headed."
Surname or Lastname
English
English : variant of Lingard.French : occupational name for a maker of or dealer in linen goods, from Old French linge ‘linen (goods)’ (see Linge 1).
Male
Greek
(ΑἰνÎας) Variant spelling of Greek AineÃas, AINEAS means "praiseworthy."
Female
Scottish
Variant spelling of Scottish Lilias, LILEAS means "lily."
Girl/Female
Indian
Dual, Second
Surname or Lastname
English
English : of uncertain origin; possibly an altered form of Irish Doyle. Compare Dyal.Name found among people of Indian origin in Guyana and Trinidad : altered spelling of Dayal. This spelling is found in Indian names occasionally when -dial is the final element of a compound personal name.
DUAL LINEAR-PROGRAM
DUAL LINEAR-PROGRAM
Boy/Male
Tamil
Light, Bright
Boy/Male
Muslim/Islamic
Discreet prudent
Girl/Female
African, American, British, Christian, English, Hebrew, Jamaican
Palm Tree; Spice; Date Palm; Beloved Angel
Boy/Male
Indian, Telugu
Lord Siva
Boy/Male
English Irish
Honey wine. Meadow.
Girl/Female
African, Arabic, Australian, Muslim, Swahili
Beauty; From Swahili
Girl/Female
Hindu, Indian
Free from Guile
Boy/Male
Tamil
Saptatala Prabhenthachha | ஸபà¯à®¤à®¾à®¤à®¾à®²à®¾ பà¯à®°à®ªà¯‡à®¨à¯à®¤à®¾à®šà®¾
Rid the curse of the seven tale trees
Boy/Male
British, English
Grinder
Girl/Female
Gujarati, Hindu, Indian, Kannada, Sanskrit, Tamil
Richness; Intelligent; Goddess Laxmi
DUAL LINEAR-PROGRAM
DUAL LINEAR-PROGRAM
DUAL LINEAR-PROGRAM
DUAL LINEAR-PROGRAM
DUAL LINEAR-PROGRAM
v. t.
To measure with a dial.
a.
Expressing, or consisting of, the number two; belonging to two; as, the dual number of nouns, etc. , in Greek.
a.
Like a line; narrow; of the same breadth throughout, except at the extremities; as, a linear leaf.
n.
One who adjusts things to a line or lines or brings them into line.
v. i.
To become dull or stupid.
n.
A part or portion; a share; hence, an indefinite quantity, degree, or extent, degree, or extent; as, a deal of time and trouble; a deal of cold.
a.
Descending in a direct line from an ancestor; hereditary; derived from ancestors; -- opposed to collateral; as, a lineal descent or a lineal descendant.
a.
In the direction of a line; of or pertaining to a line; measured on, or ascertained by, a line; linear; as, lineal magnitude.
a.
Linear.
a.
Of, pertaining to, or included by, two lines; as, bilinear coordinates.
a.
Composed of lines; delineated; as, lineal designs.
n.
One who lines, as, a liner of shoes.
n.
Specifically: To distribute, as cards, to the players at the commencement of a game; as, to deal the cards; to deal one a jack.
superl.
Furnishing little delight, spirit, or variety; uninteresting; tedious; cheerless; gloomy; melancholy; depressing; as, a dull story or sermon; a dull occupation or period; hence, cloudy; overcast; as, a dull day.
a.
Of a linear shape.
adv.
In a linear manner; with lines.
superl.
Not bright or clear to the eye; wanting in liveliness of color or luster; not vivid; obscure; dim; as, a dull fire or lamp; a dull red or yellow; a dull mirror.
a.
Of or pertaining to a line; consisting of lines; in a straight direction; lineal.
v. t.
To survey with a dial.
n.
Wood of the pine or fir; as, a floor of deal.