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A min-max optimization (MMO) problem is a mathematical optimization problem of the following form: min x ∈ R d x max y ∈ R d y f ( x , y ) such
Min-max_optimization
Optimization method
Lexicographic max-min optimization (also called lexmaxmin or leximin or leximax or lexicographic max-ordering optimization) is a kind of multi-objective
Lexicographic max-min optimization
Lexicographic_max-min_optimization
Topics referred to by the same term
Min-max may refer to: Min-max optimization Min-max theorem Min-maxing, a strategy in video games Minimax (disambiguation) Min-max heap This disambiguation
Min-max
2023 turn-based tactical video game
games, and this demographic is more likely to make in-game purchases. Jang Min-young, writing for Sisa Journal, states that the turn-based tactical game
Girls'_Frontline_2:_Exilium
Model-free reinforcement learning algorithm
Proximal policy optimization (PPO) is a reinforcement learning (RL) algorithm for training an intelligent agent. Specifically, it is a policy gradient
Proximal_policy_optimization
Principle in mathematical optimization
In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives
Duality_(optimization)
Sequential model-based optimization of expensive black-box functions
Bayesian optimization is a sequential model-based strategy for global optimization of black-box objective functions whose evaluations are costly. It is
Bayesian_optimization
approach or scenario optimization approach is a technique for obtaining solutions to robust optimization and chance-constrained optimization problems based
Scenario_optimization
Mathematical concept
Multi-objective optimization or Pareto optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, or multiattribute
Multi-objective_optimization
American computer scientist
Ye is a co-founder of minMax Optimization, a technology company based in Palo Alto and Shanghai focused on creating optimization tools for geospatial and
Yinyu_Ye
Mathematical inequality
In mathematics, the max–min inequality is as follows: For any function f : Z × W → R , {\displaystyle \ f:Z\times W\to \mathbb {R} \ ,} sup z ∈ Z
Max–min_inequality
Study of mathematical algorithms for optimization problems
generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from
Mathematical_optimization
Gives conditions that guarantee the max–min inequality holds with equality
optimization, a minimax theorem is a theorem that claims that max x ∈ X min y ∈ Y f ( x , y ) = min y ∈ Y max x ∈ X f ( x , y ) {\displaystyle \max _{x\in
Minimax_theorem
Equivalence of optimization problems
In computer science and optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source
Max-flow_min-cut_theorem
Type of multi-objective optimization
Lexicographic optimization is a kind of multi-objective optimization. In general, multi-objective optimization deals with optimization problems with two
Lexicographic_optimization
Variant of heap data structure
equal to (≥) the child keys are called max-heaps; those where it is less than or equal to (≤) are called min-heaps. Efficient (that is, logarithmic time)
Binary_heap
Economical computational problem
case of NE computation in two-player zero-sum games is known as min-max optimization. The present page studies the more general problem of non-zero-sum
Nash_equilibrium_computation
Computer science data structure
In a max heap, for any given node C, if P is the parent node of C, then the key (the value) of P is greater than or equal to the key of C. In a min heap
Heap_(data_structure)
Problem of finding the best feasible solution
science and economics, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided
Optimization_problem
Mathematical optimization theory
Robust optimization is a field of mathematical optimization theory that deals with optimization problems in which a certain measure of robustness is sought
Robust_optimization
Phenomenon in engineering
is max-min fair if and only if a data flow between any two nodes has at least one bottleneck link. Fairness measure Max-min fairness Optimization (computer
Bottleneck_(engineering)
Optimization algorithm
numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. As an example, ant colony optimization is a class
Ant colony optimization algorithms
Ant_colony_optimization_algorithms
Optimizing objective functions that have constrained variables
In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function
Constrained_optimization
Skeletonized version of algebraic geometry
their solutions have important applications in optimization problems, for example, the problem of optimizing departure times for a network of trains. Tropical
Tropical_geometry
Inputs at which function values are highest
the arguments of the maxima (abbreviated arg max or argmax) and arguments of the minima (abbreviated arg min or argmin) are the input points at which a
Arg_max
Optimization algorithm
In mathematics, the spiral optimization (SPO) algorithm is a metaheuristic inspired by spiral phenomena in nature. The first SPO algorithm was proposed
Spiral_optimization_algorithm
Class of reinforcement learning algorithms
sub-class of policy optimization methods. Unlike value-based methods which learn a value function to derive a policy, policy optimization methods directly
Policy_gradient_method
Measure of the decline from a historical peak
( T ) = max τ ∈ ( 0 , T ) D ( τ ) = max τ ∈ ( 0 , T ) [ max t ∈ ( 0 , τ ) X ( t ) − X ( τ ) ] {\displaystyle \operatorname {MDD} (T)=\max _{\tau \in
Drawdown_(economics)
Population-based search algorithm
global search, and can be used for both combinatorial optimization and continuous optimization. The only condition for the application of the bees algorithm
Bees_algorithm
Optimization algorithm
already been introduced, and was added to SGD optimization techniques in 1986. However, these optimization techniques assumed constant hyperparameters,
Stochastic_gradient_descent
Computational problem in graph theory
s-t cut (i.e., cut severing s from t) in the network, as stated in the max-flow min-cut theorem. The maximum flow problem was first formulated in 1954 by
Maximum_flow_problem
Topics referred to by the same term
(Australian TV series) Max-Min Ant System, a type of Ant colony optimization algorithm, see Ant colony optimization algorithms#Max-Min Ant System (MMAS) Search
MMAS
Machine learning technique
function to improve an agent's policy through an optimization algorithm like proximal policy optimization. RLHF has applications in various domains in machine
Reinforcement learning from human feedback
Reinforcement_learning_from_human_feedback
Non-probabilistic decision-making model
statistics, robust optimization, operations research, philosophy, etc. One of the most famous examples of a Maximin/Minimax model is min x ∈ R max y ∈ R { x
Wald's_maximin_model
Decision-making strategy
2,...,n; fi* = max (fij,j=1,...,J), fi^ = min (fij,j=1,...,J), if the i-th function is benefit; fi* = min (fij,j=1,...,J), fi^ = max (fij,j=1,...,J)
VIKOR_method
Problem in graph theory
problem. The canonical optimization variant of the above decision problem is usually known as the Maximum-Cut Problem or Max-Cut and is defined as: Given
Maximum_cut
programs. SIAM J. on Optimization, 19(3), 1211-1230, 2008.[3] A. Carè, S. Garatti and M.C. Campi. Scenario min-max optimization and the risk of empirical
Marco_Claudio_Campi
Mathematical propositions in network flow theory
theory, approximate max-flow min-cut theorems concern the relationship between the maximum flow rate (max-flow) and the minimum cut (min-cut) in multi-commodity
Approximate max-flow min-cut theorem
Approximate_max-flow_min-cut_theorem
1957 technique for modelling problems of decision making under uncertainty
)} f 2 ( 0 ) = min { b success probability in periods 2,3,4 max 0 0.4 ( 0 ) + 0.6 ( 0 ) = 0 ← b 2 ( 0 ) = 0 {\displaystyle f_{2}(0)=\min
Stochastic dynamic programming
Stochastic_dynamic_programming
Class of computational problems
the flow amounts are restricted to a finite set of nonzero values The max-flow min-cut theorem equates the value of a maximum flow to the value of a minimum
Network_flow_problem
Optimization for dynamical systems
Lyapunov optimization for dynamical systems. It gives an example application to optimal control in queueing networks. Lyapunov optimization refers to
Lyapunov_optimization
Mathematical optimization problem
as large as possible, such that the sum in each subset j is at most Cj. Max-min MSSP (also called bottleneck MSSP or BMSSP): again each subset has a capacity
Multiple_subset_sum
Advanced method of process control
computational cost. The basic idea behind the min/max MPC approach is to modify the on-line "min" optimization to a "min-max" problem, minimizing the worst case
Model_predictive_control
Subfield of convex optimization
field of optimization which is of growing interest for several reasons. Many practical problems in operations research and combinatorial optimization can be
Semidefinite_programming
American computer scientist and educator
algorithms and optimization problems involving graphs, cuts, flows, and embeddings. With Leighton, he developed multicommodity max-flow min-cut theorems
Satish_B._Rao
Mathematical approximation
α → max {\displaystyle m_{\alpha }\to \max } as α → ∞ {\displaystyle \alpha \to \infty } and m α → min {\displaystyle m_{\alpha }\to \min }
Smooth_maximum
employed for MRF optimization. Dual decomposition is applied to markov logic programs as an inference technique. Discrete MRF Optimization (inference) is
MRF optimization via dual decomposition
MRF_optimization_via_dual_decomposition
Partition of a graph by removing fewest possible edges
source side of the cut to the sink side of the cut. As shown in the max-flow min-cut theorem, the weight of this cut equals the maximum amount of flow
Minimum_cut
Smooth approximation to the maximum function
to the maximum max i x i {\displaystyle \max _{i}x_{i}} with the following bounds max { x 1 , … , x n } ≤ L S E ( x 1 , … , x n ) ≤ max { x 1 , … , x n
LogSumExp
Second-order deterministic global optimization algorithm
global optimization approach for Lennard-Jones microclusters." Journal of Chemical Physics, 1992, 97(10), 7667-7677 "αBB: A global optimization method
ΑΒΒ
Mathematical function with convex lower level sets
mathematical analysis, in mathematical optimization, and in game theory and economics. In nonlinear optimization, quasiconvex programming studies iterative
Quasiconvex_function
Problem optimization method
sub-problems. In the optimization literature, this relationship is called the Bellman equation. In terms of mathematical optimization, dynamic programming
Dynamic_programming
Optimization technique
cuts" is applied specifically to those models which employ a max-flow/min-cut optimization (other graph cutting algorithms may be considered as graph partitioning
Graph cuts in computer vision and artificial intelligence
Graph_cuts_in_computer_vision_and_artificial_intelligence
Framework for modeling optimization problems that involve uncertainty
In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. A stochastic
Stochastic_programming
Matrix approximation problem in linear algebra
_{F}\\&=\arg \min _{\Omega }\|A\|_{F}^{2}+\|B\|_{F}^{2}-2\langle \Omega A,B\rangle _{F}\\&=\arg \max _{\Omega }\langle \Omega A,B\rangle _{F}\\&=\arg \max _{\Omega
Orthogonal_Procrustes_problem
Java math library
10); // precision, max number of iterations UnivariateMinimizer.Solution soln = solver.solve(logGamma); // optimization double x_min = soln.search(0, 5);
SuanShu_numerical_library
Rawlsian decision rule for social choice
that is, it solves the following optimization problem: max x ∈ X min i ∈ I u i ( x ) . {\displaystyle \max _{x\in X}\min _{i\in I}u_{i}(x).} Often, there
Egalitarian_rule
Combinatorial optimization problem
unconstrained binary optimization (QUBO), also known as unconstrained binary quadratic programming (UBQP), is a combinatorial optimization problem with a wide
Quadratic unconstrained binary optimization
Quadratic_unconstrained_binary_optimization
Complexity class of approximable problems
approximability complexity classes MaxSNP - a closely related subclass Complexity Zoo: APX C. Papadimitriou and M. Yannakakis. Optimization, approximation and complexity
APX
Largest and smallest value taken by a function at a given point
the arguments of the maxima (abbreviated arg max or argmax) and arguments of the minima (abbreviated arg min or argmin) are the input points at which a
Maximum_and_minimum
2024 smartphone by Apple
The iPhone 16 Pro and iPhone 16 Pro Max are smartphones developed and marketed by Apple. Alongside the iPhone 16 and iPhone 16 Plus, they form the eighteenth
IPhone_16_Pro
Search algorithm or heuristic method to solve constraint satisfaction problems
science, a min-conflicts algorithm is a search algorithm or heuristic method to solve constraint satisfaction problems. One such algorithm is min-conflicts
Min-conflicts_algorithm
Mathematical model to assist inventory levels
D min = 50 {\displaystyle D_{\min }=50} and D max = 80 {\displaystyle D_{\max }=80} . q opt = F − 1 ( 7 − 5 7 ) = F − 1 ( 0.285 ) = D min + ( D max −
Newsvendor_model
the order of the min max to max min (see the references for more details), the optimization problem can be formulated as: R C C = max ( Δ , x ) ∈ T { −
Chebyshev_center
Software engineering technique
one of them with a semantics-preserving optimization. Global value numbering (GVN) is a compiler optimization based on the static single-assignment form
Value_numbering
Science, Optimal Computing Budget Allocation (OCBA) is a simulation optimization method designed to maximize the Probability of Correct Selection (PCS)
Optimal computing budget allocation
Optimal_computing_budget_allocation
Genetic operation used to add population diversity
min , x max ] {\displaystyle [x_{\min },x_{\max }]} of the gene to be changed, e.g.: σ = x max − x min 6 {\displaystyle \sigma ={\frac {x_{\text{max
Mutation (evolutionary algorithm)
Mutation_(evolutionary_algorithm)
mathematical optimization, total dual integrality is a sufficient condition for the integrality of a polyhedron. Thus, the optimization of a linear objective
Total_dual_integrality
Biogeography-based optimization (BBO) is an evolutionary algorithm (EA) that optimizes a function by stochastically and iteratively improving candidate
Biogeography-based optimization
Biogeography-based_optimization
Constrained least squares problem
In mathematical optimization, the problem of non-negative least squares (NNLS) is a type of constrained least squares problem where the coefficients are
Non-negative_least_squares
domains)); } // Find min and max cost genes boolean[] minGene = null, maxGene = null; double minCost = POSITIVE_INFINITY, maxCost = NEGATIVE_INFINITY;
Population-based incremental learning
Population-based_incremental_learning
Type of algorithm for constrained optimization
In mathematical optimization, penalty methods are a certain class of algorithms for solving constrained optimization problems. A penalty method replaces
Penalty_method
In mathematics, the map segmentation problem is a kind of optimization problem. It involves a certain geographic region that has to be partitioned into
Map_segmentation
General-purpose MATLAB software
applications such as performance optimization of Formula One race cars, Ref. where the software has been used for minimum-time optimization of low-thrust orbital
GPOPS-II
Modelling of electrical grids
expansion optimization Transmission expansion optimization Generation-transmission expansion co-optimization Distribution network optimization A well-defined
Power_system_simulation
Optimization algorithm
Frank–Wolfe algorithm is an iterative first-order optimization algorithm for constrained convex optimization. Also known as the conditional gradient method
Frank–Wolfe_algorithm
hence the ellipsoid method can usually be computed in polynomial time. Nemirovsky and Ben-Tal (2023). "Optimization III: Convex Optimization" (PDF).
Center-of-gravity_method
Data structure in tree form sorted for fast lookup
EMPTY_TREE min := node while min.left is not NULL min := min.left return min.key findMaximum(node) if node is NULL return EMPTY_TREE max := node while max.right
Search_tree
Class of algorithms that find approximate solutions to optimization problems
algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on
Approximation_algorithm
Mathematical Theory
mathematical theory of probability, the drift-plus-penalty method is used for optimization of queueing networks and other stochastic systems. The technique is for
Drift_plus_penalty
Concept in mathematics
Zlobec S. (2009). Nondifferentiable optimization: Parametric programming. Pp. 2607-2615, in Encyclopedia of Optimization, Floudas C.A and Pardalos, P.M. editors
Stability_radius
Generalization of binary functions
In mathematics and optimization, a pseudo-Boolean function is a function of the form f : B n → R , {\displaystyle f:\mathbf {B} ^{n}\to \mathbb {R} ,}
Pseudo-Boolean_function
Optimization algorithms using quantum computing
Quantum optimization algorithms are quantum algorithms that are used to solve optimization problems. Mathematical optimization deals with finding the best
Quantum optimization algorithms
Quantum_optimization_algorithms
Mathematical optimization problem restricted to integers
An integer programming, also known as integer optimization, problem is a mathematical optimization or feasibility program in which some or all of the variables
Integer_programming
Method to solve optimization problems
programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject
Linear_programming
Set-to-real map with diminishing returns
(2003), Combinatorial Optimization, Springer, ISBN 3-540-44389-4 Lee, Jon (2004), A First Course in Combinatorial Optimization, Cambridge University Press
Submodular_set_function
Machine learning framework
main components. One is casting optimization into a game, of form min G max D L ( G , D ) {\displaystyle \min _{G}\max _{D}L(G,D)} , which is different
Generative adversarial network
Generative_adversarial_network
Shared independent set of two matroids
have max I ∈ I 1 ∩ I 2 | I | = min A ⊆ E ( r 1 ( A ) + r 2 ( E ∖ A ) ) {\displaystyle \max _{I\in {\mathcal {I}}_{1}\cap {\mathcal {I}}_{2}}|I|=\min _{A\subseteq
Matroid_intersection
Theorem in convex analysis
the form f ( x ) = max z ∈ Z ϕ ( x , z ) . {\displaystyle f(x)=\max _{z\in Z}\phi (x,z).} The theorem has applications in optimization, where it sometimes
Danskin's_theorem
Internet slang suffix meaning to optimize or maximize something
role-playing games as "min-maxing", a strategy where a player dumps all available resources into a single tactic, optimizing for raw mechanical advantage
-maxxing
Primal-Dual algorithm optimization for convex problems
mathematics, the Chambolle–Pock algorithm is an algorithm used to solve convex optimization problems. It was introduced by Antonin Chambolle and Thomas Pock in 2011
Chambolle–Pock_algorithm
Metaheuristic method for optimization problems
metaheuristic method for solving a set of combinatorial optimization and global optimization problems. It explores distant neighborhoods of the current
Variable_neighborhood_search
Fair item allocation problem
Egalitarian item allocation, also called max-min item allocation is a fair item allocation problem, in which the fairness criterion follows the egalitarian
Egalitarian_item_allocation
American/Canadian mathematician and computer scientist
Combinatorics and Optimization at the University of Waterloo's Faculty of Mathematics where his research encompassed combinatorial optimization problems and
Jack_Edmonds
Generalization of the Legendre transformation
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex
Convex_conjugate
Variation of minimax game tree search
This algorithm relies on the fact that min ( a , b ) = − max ( − b , − a ) {\displaystyle \min(a,b)=-\max(-b,-a)} to simplify the implementation of
Negamax
Optimization and sampling technique
(SGLD) is an optimization and sampling technique composed of characteristics from Stochastic gradient descent, a Robbins–Monro optimization algorithm, and
Stochastic gradient Langevin dynamics
Stochastic_gradient_Langevin_dynamics
High-end premium Android smartphones
The Xiaomi 17 Pro and Xiaomi 17 Pro Max are high-end premium Android smartphones developed, designed, and marketed by Chinese electronic consumer Xiaomi
Xiaomi_17_Pro
dynamic optimization of distillation operations, Computers & Chemical Engineering, 28 (10) (2004) 2037-2052. MPEC examples such as SIGN, ABS, MIN, and MAX Formulating
Mathematical programming with equilibrium constraints
Mathematical_programming_with_equilibrium_constraints
Fair division protocol in computing
maximum x can be found by solving a linear program; see Lexicographic max-min optimization. Alternatively, the DRF can be computed sequentially. The algorithm
Dominant_resource_fairness
travel, tourism, insurance
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MIN MAX-OPTIMIZATION
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travel, tourism, insurance