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Concept in integral mathematics
the solution to the original integer program. Consider the set cover problem, the linear programming relaxation of which was first considered by Lovász
Linear_programming_relaxation
information about the original problem. For example, a linear programming relaxation of an integer programming problem removes the integrality constraint and
Relaxation_(approximation)
Mathematical optimization problem restricted to integers
integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear. Integer programming is
Integer_programming
Method to solve optimization problems
and objective are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization)
Linear_programming
Graph coloring where graph elements are assigned sets of colors
the linear programming relaxation of traditional graph coloring. Indeed, fractional coloring problems are much more amenable to a linear programming approach
Fractional_coloring
Combinatorial optimization method
cutting planes to tighten the linear programming relaxations. Note that if cuts are only used to tighten the initial LP relaxation, the algorithm is called
Branch_and_cut
Iterative solving method
solving nonlinear systems of equations. Relaxation methods are important especially in the solution of linear systems used to model elliptic partial differential
Relaxation_(iterative_method)
Method of solving a linear system of equations
In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations
Successive_over-relaxation
Optimization problem in mathematics
second-order cone programming (SOCP) and linear programming (LP) relaxations providing the same objective value as the SDP relaxation are available. Nonconvex
Quadratically constrained quadratic program
Quadratically_constrained_quadratic_program
Mathematical optimization concept
(optimization) Semidefinite programming Relaxation (approximation) Gärtner, Bernd; Matoušek, Jiří (2006). Understanding and Using Linear Programming. Berlin: Springer
Dual_linear_program
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
Mathematical combinatorial optimization method
added to the linear programming relaxation (LP relaxation). At the start of the algorithm, sets of columns are excluded from the LP relaxation in order to
Branch_and_price
Concept in computational geometry
pseudo-disks-set with n objects and union complexity u. Using linear programming relaxation, it is possible to find a disjoint set of size at least n u
Maximum_disjoint_set
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
Subset of a graph's vertices, including at least one endpoint of every edge
algorithm for the minimum vertex cover problem. Furthermore, the linear programming relaxation of that ILP is half-integral, that is, there exists an optimal
Vertex_cover
On short connecting nets with added points
{\displaystyle \ln(4)+\varepsilon \leq 1.39} approximation using a linear programming relaxation and a technique called iterative, randomized rounding. The general
Steiner_tree_problem
Belgian-American mathematician
Fulkerson Prize for joint work with David P. Williamson on the semidefinite programming approximation algorithm for the maximum cut problem. In 2012 Goemans
Michel_Goemans
Method in mathematical optimization
Lagrangian Relaxation for mixed-integer linear programming," Scientific Reports. 12: 22417, doi:10.1038/s41598-022-26264-1 Neal Young, Lagrangian Relaxation Example
Lagrangian_relaxation
constrained quadratic program Linear-fractional programming — objective is ratio of linear functions, constraints are linear Fractional programming — objective
List of numerical analysis topics
List_of_numerical_analysis_topics
Subfield of mathematical optimization
transformations: Linear programming problems are the simplest convex programs. In LP, the objective and constraint functions are all linear. Quadratic programming are
Convex_optimization
Principle in mathematical optimization
primal and dual programs together is often easier than solving only one of them. Examples are linear programming and quadratic programming. A better and
Duality_(optimization)
Class of problems in computer science
approximation for the weighted case. Using the technique of Linear programming relaxation, it is possible to approximate the optimal scheduling with slightly
Interval_scheduling
Computational problem of graph theory
total delay being at most the delay budget. Fractional RSP is a linear programming relaxation of RSP. Handler and Zang introduced it and gave a combinatorial
Shortest_path_problem
Iterative method used to solve a linear system of equations
over-relaxation Iterative method § Linear systems Gaussian Belief Propagation Matrix splitting Saad, Yousef (2003). Iterative Methods for Sparse Linear Systems
Jacobi_method
Classical problem in combinatorics
solution of the linear programming relaxation. Let x S ∗ {\displaystyle {x_{S}^{*}}} be an optimal fractional solution to the LP relaxation. Each set S ∈
Set_cover_problem
Class of algorithms that find approximate solutions to optimization problems
mathematical programming formulation (typically a convex programming) such as Linear programming, Semidefinite programming, etc, to obtain a relaxation of solutions
Approximation_algorithm
Cycle graph with all opposite nodes linked
problems can be used to define facets of the polytope describing a linear programming relaxation of the problem; these facets are called Möbius ladder constraints
Möbius_ladder
branch-and-bound where every subproblem is solved by constructing a linear programming relaxation to obtain a lower bound. Branching may occur at both continuous
Couenne
Mathematical optimization software
integer programming problem by a branch and bound algorithm with linear programming relaxations. It also provides automatic constraint classification, preprocessing
MINTO
Edges that hit all cycles in a graph
scheme. Its main ideas are to apply randomized rounding to a linear programming relaxation of the problem, and to derandomize the resulting algorithm using
Feedback_arc_set
Decay of nuclear spin polarization in MRI and NMR
equilibrium value is termed spin-lattice relaxation while the loss of phase-coherence of the spins is termed spin-spin relaxation, which is manifest as an observed
Relaxation_(NMR)
Class of statistical modeling methods
Loopy belief propagation Alpha expansion Mean field inference Linear programming relaxations Learning the parameters θ {\displaystyle \theta } is usually
Conditional_random_field
Computer programming for quantum computers
develop functional programming languages for quantum computing. Functional programming languages are well-suited for reasoning about programs. Examples include
Quantum_programming
Rational design of new protein molecules
large instances of protein design problems. These solvers use a linear programming relaxation of the problem, where qi and qij are allowed to take continuous
Protein_design
Magnetic phenomenon
In physics, the spin–spin relaxation is the mechanism by which Mxy, the transverse component of the magnetization vector, exponentially decays towards
Spin–spin_relaxation
mathematics, Graver bases enable iterative solutions of linear and various nonlinear integer programming problems in polynomial time. They were introduced by
Graver_basis
and superdiagonals. Linear independence — two or more vectors are linearly independent if there is no way to construct one from linear combinations of the
List_of_named_matrices
Type of electronic circuit
general types of electronic oscillators: the linear or harmonic oscillator, and the nonlinear or relaxation oscillator. The two types are fundamentally
Electronic_oscillator
trade-offs between relaxation quality and computational efficiency. The symmetric formulation improves the linear programming relaxation by distinguishing
Maximum_common_edge_subgraph
Combinatorial optimization problem
generalized assignment problem is NP-hard. However, there are linear-programming relaxations which give a ( 1 − 1 / e ) {\displaystyle (1-1/e)} -approximation
Generalized assignment problem
Generalized_assignment_problem
Optimization technique for solving (mixed) integer linear programs
by solving a non-integer linear program, the linear relaxation of the given integer program. The theory of Linear Programming dictates that under mild
Cutting-plane_method
solved as an integer linear program (ILP). Compute an optimal fractional solution x {\displaystyle x} to the linear programming relaxation (LP) of the ILP
Randomized_rounding
Study of mathematical algorithms for optimization problems
mathematical programming problem (a term not directly related to computer programming, but still in use for example in linear programming – see History
Mathematical_optimization
Physical phenomenon
During nuclear magnetic resonance observations, spin–lattice relaxation is the mechanism by which the longitudinal component of the total nuclear magnetic
Spin–lattice_relaxation
Numerical approximation algorithm
U\right)\quad (\omega \not \in \{0,2\})} Linear stationary iterative methods are also called relaxation methods. Krylov subspace methods work by forming
Iterative_method
Optimization problem
performs significantly better, in the sense that it has a tighter linear programming relaxation than the first formulation. Notice that summing the new constraints
Optimal_facility_location
Iterative method used to solve a linear system of equations
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method
Gauss–Seidel_method
American operations researcher (born 1937)
linear programming relaxation as well as some of the nodes that have a value of 0.5. Nemhauser is the author of Introduction to Dynamic Programming (Wiley
George_Nemhauser
Computational benchmark
This computing paradigm based upon sending identical photons through a linear-optical network can solve certain sampling and search problems that, assuming
Quantum_supremacy
Any method, process, procedure, or activity that helps a person to relax
Additionally, there was a linear association between progressive muscle relaxation & guided imagery and physiological relaxation, while the deep breathing
Relaxation_technique
Algorithm for solving linear programming problems with special structure
the tractability of large-scale linear programs or create a tighter linear relaxation of mixed integer linear programs. The Dantzig-Wolfe decomposition
Dantzig–Wolfe_decomposition
Computer scientist
subproblems, an efficient solution was attained using a partial linear programming relaxation algorithm. Furthermore, he conducted an extensive review of
George_N._Rouskas
_{0}} problem. Note that this relaxation is convex and hence amenable to the standard techniques of linear programming - a computationally desirable feature
Nullspace_property
Paradigm of quantum computer
Linear optical quantum computing or linear optics quantum computation (LOQC), also photonic quantum computing (PQC), is a paradigm of quantum computation
Linear optical quantum computing
Linear_optical_quantum_computing
2011 book by William J. Cook
solving the problem, leading from heuristics and metaheuristics, linear programming relaxation, and cutting-plane methods, up to the branch and bound method
In Pursuit of the Traveling Salesman
In_Pursuit_of_the_Traveling_Salesman
Theorem in physics
energy, spin — are represented by "observables", which are self-adjoint linear operators acting on the Hilbert space. When an observable is measured, the
Bell's_theorem
Principle in quantum information theory
supremacy Quantum volume QC scaling laws Randomized benchmarking XEB Relaxation times T1 T2 Quantum computing models Adiabatic quantum computation Continuous-variable
No-communication_theorem
Proposed quantum computer implementation
configuration in z ^ {\displaystyle {\widehat {z}}} , the simplest case being a linear strand of only a few ions. Coulomb interactions of increasing complexity
Trapped-ion_quantum_computer
American mathematician
fractional solution of a linear programming relaxation and using the properties of the optimal solutions of the linear program and a generalization of
David_Shmoys
Award
parallel computers". 1991: Michel Goemans for "Analysis of Linear Programming Relaxations for a Class of Connectivity Problems". Other Finalists: Leslie
Tucker_Prize
Quantum search algorithm
Implementing the steps for this algorithm can be done using a number of gates linear in the number of qubits. Thus, the gate complexity of this algorithm is
Grover's_algorithm
Algorithm to be run on quantum computers
faster than the best possible classical algorithm for the same task, a linear search. Quantum algorithms are usually described, in the commonly used circuit
Quantum_algorithm
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
constraints can be thought of as the fractional solutions of a linear programming relaxation of the stable matching problem. It is a theorem of Vande Vate
Stable_matching_polytope
Upper bound on the knowable information of a quantum state
supremacy Quantum volume QC scaling laws Randomized benchmarking XEB Relaxation times T1 T2 Quantum computing models Adiabatic quantum computation Continuous-variable
Holevo's_theorem
Quantum physics-based metaheuristic for optimization problems
doi:10.1038/nature10012. PMID 21562559. S2CID 205224761. "Learning to program the D-Wave One". D-Wave Systems blog. Archived from the original on July
Quantum_annealing
when relaxation of the feasibility region is used. The inscribed form of the Chebyshev center problem can be formulated as a linear programming problem
Chebyshev_center
Algorithmic technique
achieves this improved bound exploits the half-integrality of the linear program relaxation of vertex cover due to Nemhauser and Trotter. Another kernelization
Kernelization
Statistical analysis technique
framework, a penalized matrix decomposition framework, a convex relaxation/semidefinite programming framework, a generalized power method framework an alternating
Sparse_PCA
Basic unit of quantum information
circular polarization) can also be measured as horizontal and vertical linear polarization. In a classical system, a bit would have to be in one state
Qubit
Process in quantum computing
tendency to relax toward thermal equilibrium and are characterized by a relaxation time. Moreover, even an isolated qubit possesses an intrinsic Hamiltonian
Quantum_error_correction
Fair item allocation problem
from rounding a suitable linear programming relaxation of the problem, and is the best possible result for this linear program. He also gave an O ( n )
Egalitarian_item_allocation
procedure or SIP, is an algorithm for solving a sparse linear system of equations Successive over-relaxation (SOR): method used to speed up convergence of the
List_of_algorithms
Type of quantum computer
twists (logic circuits) to the topological quantum computer, in a simple linear relationship. In other words, a reasonable increase in elements (braid twists)
Topological_quantum_computer
at most k. They show that the linear-program relaxation of this variant has the same optimal value as the LP relaxation of the unconstrained variant.
Balanced_number_partitioning
Numerical optimization process
optimization is also known as the Lasserre hierarchy of semidefinite programming relaxations. Sum-of-squares optimization techniques have been applied across
Sum-of-squares_optimization
Deterministic quantum algorithm
supremacy Quantum volume QC scaling laws Randomized benchmarking XEB Relaxation times T1 T2 Quantum computing models Adiabatic quantum computation Continuous-variable
Deutsch–Jozsa_algorithm
Topological quantum error correcting code
Daniel; Herdman, C. M.; Gorman, D. J.; Whaley, K. B. (7 October 2014). "Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size
Surface_code
Quantum algorithm for integer factorization
PMID 23846653. Bernstein, Daniel (1998). "Detecting perfect powers in essentially linear time". Mathematics of Computation. 67 (223): 1253–1283. doi:10.1090/S0025-5718-98-00952-1
Shor's_algorithm
regularity conditions, equal to the value of the convex relaxation of the primal problem: The convex relaxation is the problem arising replacing a non-convex feasible
Duality_gap
Change of basis applied in quantum computing
In quantum computing, the quantum Fourier transform (QFT) is a linear transformation on quantum bits, and is the quantum analogue of the discrete Fourier
Quantum_Fourier_transform
Computer hardware technology that uses quantum mechanics
bit, which can be in one of two states (a binary), a qubit can exist in a linear combination of states known as a quantum superposition. The result of measuring
Quantum_computing
General-purpose programming language
high-level general-purpose programming language that supports both object-oriented programming and functional programming. Designed to be concise, many
Scala_(programming_language)
Foundational object in quantum communication theory
system B. However, once a linear map Φ {\displaystyle \Phi } between the density matrices is specified, a standard linearity argument, together with the
Quantum_channel
Branch of numerical optimization
optimality. Linear programming optimization problems strictly fall under the category of deterministic global optimization. Much like linear programming problems
Deterministic global optimization
Deterministic_global_optimization
Quantum algorithm for eigenvalue estimation
quantum algorithms, such as Shor's algorithm, the quantum algorithm for linear systems of equations, and the quantum counting algorithm. The algorithm
Quantum phase estimation algorithm
Quantum_phase_estimation_algorithm
Search problem in quantum mechanics
The hidden linear function problem, is a search problem that generalizes the Bernstein–Vazirani problem. In the Bernstein–Vazirani problem, the hidden
Hidden linear function problem
Hidden_linear_function_problem
Methods for numerical approximations
instance, linear programming deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the
Numerical_analysis
Quantum key distribution protocol
supremacy Quantum volume QC scaling laws Randomized benchmarking XEB Relaxation times T1 T2 Quantum computing models Adiabatic quantum computation Continuous-variable
BB84
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
Restricted model of non-universal quantum computation
sampling from the probability distribution of identical bosons scattered by a linear interferometer. Although the problem is well defined for any bosonic particles
Boson_sampling
segments, side-chains and branches. The linearity of the sf-TM curve will be changed by such transitions. Other relaxations may be due to release of internal
Thermomechanical_analysis
Interdisciplinary research area
University of Berlin in Germany. Differentiable programming Quantum computing Quantum algorithm for linear systems of equations Quantum annealing Quantum
Quantum_machine_learning
Computational complexity class of problems
supremacy Quantum volume QC scaling laws Randomized benchmarking XEB Relaxation times T1 T2 Quantum computing models Adiabatic quantum computation Continuous-variable
BQP
Linear optical quantum computing implementation
The KLM scheme or KLM protocol is an implementation of linear optical quantum computing (LOQC) developed in 2000 by Emanuel Knill, Raymond Laflamme and
KLM_protocol
Any algorithm which solves the search problem
exploit partial knowledge about the structure of this space, such as linear relaxation, constraint generation, and constraint propagation. An important subclass
Search_algorithm
Chinese-American mathematician
for structured convex programs and network flow problems, Complexity analysis of interior point methods for linear programming, Parallel and distributed
Paul_Tseng
Transforms equations for numerical solution
preconditioned problem is then usually solved by an iterative method. In linear algebra and numerical analysis, a preconditioner P {\displaystyle P} of
Preconditioner
Secure communication method
transmit two messages by encoding them in two "conjugate observables", such as linear and circular polarization of light, so that either, but not both, of which
Quantum_key_distribution
Theorem in quantum information science
supremacy Quantum volume QC scaling laws Randomized benchmarking XEB Relaxation times T1 T2 Quantum computing models Adiabatic quantum computation Continuous-variable
No-cloning_theorem
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