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Functions used to evaluate optimization algorithms
objective functions for single-objective optimization cases are presented. In the second part, test functions with their respective Pareto frontiers for multi-objective
Test functions for optimization
Test_functions_for_optimization
Function used as a performance test problem for optimization algorithms
mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem
Rosenbrock_function
Study of mathematical algorithms for optimization problems
Mathematical optimization algorithms Mathematical optimization software Process optimization Simulation-based optimization Test functions for optimization Vehicle
Mathematical_optimization
Function used as a performance test problem for optimization algorithms
In mathematical optimization, the Ackley function is a non-convex function used as a performance test problem for optimization algorithms. It was proposed
Ackley_function
mathematics, the Griewank test function is a smooth multidimensional mathematical function used in unconstrained optimization. It is commonly employed
Griewank_function
Function used as a performance test problem for optimization algorithms
for up to n = 10 {\displaystyle n=10} . Test functions for optimization Molga, M.; Smutnicki, C. (2005). "Test functions for optimization needs. Test
Shekel_function
Function used as a performance test problem for optimization algorithms
Rastrigin function of two variables In mathematical optimization, the Rastrigin function is a non-convex function used as a performance test problem for optimization
Rastrigin_function
Function used as a performance test problem for optimization algorithms
Himmelblau's function In mathematical optimization, Himmelblau's function is a multi-modal function, used to test the performance of optimization algorithms
Himmelblau's_function
Optimization performance test
mathematical optimization, the sphere function is a convex function used as a performance test problem for optimization algorithms. The sphere function was proposed
Sphere_function
types of functions Test functions for optimization List of mathematical abbreviations List of special functions and eponyms Special functions[link removed] :
List of mathematical functions
List_of_mathematical_functions
Objective function of evolutionary algorithm
also used in other metaheuristics, such as ant colony optimization or particle swarm optimization. In the field of EAs, each candidate solution, also called
Fitness_function
Process of finding the optimal set of variables for a machine learning algorithm
hyperparameter optimization methods. Bayesian optimization is a global optimization method for noisy black-box functions. Applied to hyperparameter optimization, Bayesian
Hyperparameter_optimization
Trial and error problem solvers with a metaheuristic or stochastic optimization character
organism simulators Mutation testing No free lunch in search and optimization Program synthesis Test functions for optimization Unconventional computing Universal
Evolutionary_computation
Topics referred to by the same term
1967 children's novel by Ivan Southall Shekel's foxholes, a test function for optimization Foxhole conversion, an aphorism used to argue that in times
Foxhole
Mathematical discipline
Derivative-free optimization (sometimes referred to as blackbox optimization) is a discipline in mathematical optimization that does not use derivative
Derivative-free_optimization
Program function without side effects
return cache[n]; } Functions that have just the above property 2 – that is, have no side effects – allow for compiler optimization techniques such as
Pure_function
Computer program optimization method
substituted. The compiler will then try to optimize the result. Whole program optimization (WPO) is the compiler optimization of a program using information about
Interprocedural_optimization
Numerical optimization algorithm
objective function in a multidimensional space. It is a direct search method (based on function comparison) and is often applied to nonlinear optimization problems
Nelder–Mead_method
Improving the efficiency of software
In computer science, program optimization, code optimization, or software optimization is the process of modifying a software system to make some aspect
Program_optimization
same function f, or a given optimization software can be used for different functions f. The following tables provide a list of notable optimization software
List_of_optimization_software
Optimization method
Stochastic optimization (SO) are optimization methods that generate and use random variables. For stochastic optimization problems, the objective functions or
Stochastic_optimization
Matrix of second derivatives
A bordered Hessian is used for the second-derivative test in certain constrained optimization problems. Given the function f {\displaystyle f} considered
Hessian_matrix
Iterative simulation method
by using another overlaying optimizer, a concept known as meta-optimization, or even fine-tuned during the optimization, e.g., by means of fuzzy logic
Particle_swarm_optimization
Mathematical relation assigning a probability event to a cost
mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event
Loss_function
Compiler that optimizes generated code
equivalent code optimized for some aspect. Optimization is limited by a number of factors. Theoretical analysis indicates that some optimization problems are
Optimizing_compiler
Practice and strategies of increasing online visibility
developed new optimization approaches for LLM-based search, referred to as answer engine optimization (AEO) or generative engine optimization (GEO). These
Search_engine_optimization
concepts: Barrier function Penalty method Trust region Test functions for optimization: Rosenbrock function — two-dimensional function with a banana-shaped
List of numerical analysis topics
List_of_numerical_analysis_topics
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
Method for finding the extrema of a function
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local
Derivative_test
Optimization algorithm
method for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. As an example, ant colony optimization is
Ant colony optimization algorithms
Ant_colony_optimization_algorithms
Real function with secant line between points above the graph itself
number). Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where
Convex_function
different optimization software modules can be easily tested on the same function f, or a given optimization software can be used for different functions f.
Comparison of optimization software
Comparison_of_optimization_software
Artificial intelligence concept
(Dynamic Reliability Adjustment for Multi-objective Optimization). This framework is based on multi-objective optimization and prediction reliability, thus
Reward_hacking
Optimization replacing a function call with that function's source code
further optimizations and improved scheduling, due to increasing the size of the function body, as better optimization is possible on larger functions. The
Inline_expansion
Fundamental analysis
data used in price optimization can include survey data, operating costs, inventories, and historic prices and sales. Price optimization practice has been
Price_optimization
Method to solve optimization problems
as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear
Linear_programming
Experiment methodology
A/B testing (also known as bucket testing, split-run testing or split testing) is a user-experience research method. A/B tests consist of a randomized
A/B_testing
Concept in mathematical optimization
mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called
Karush–Kuhn–Tucker_conditions
Optimization by removing non-optimal solutions to subproblems
an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm
Branch_and_bound
Checking software against expectations
points have been tested. Code coverage as a software metric can be reported as a percentage for: Function coverage, which reports on functions executed Statement
Software_testing
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
Simulation-based optimization (also known as simply simulation optimization) integrates optimization techniques into simulation modeling and analysis
Simulation-based_optimization
Finding multiple solutions of a problem
resulting in their global optimization ability on multimodal functions. Moreover, the techniques for multimodal optimization are usually borrowed as diversity
Evolutionary multimodal optimization
Evolutionary_multimodal_optimization
Optimization algorithm
gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable
Stochastic_gradient_descent
Application of metaheuristic search techniques to software engineering
as optimization problems. Optimization techniques of operations research such as linear programming or dynamic programming are often impractical for large
Search-based software engineering
Search-based_software_engineering
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
first-order optimization algorithm for constrained convex optimization Golden-section search: an algorithm for finding the maximum of a real function Gradient
List_of_algorithms
Mathematical folklore
easy shortcuts to designing optimization algorithms. It appeared in the 1997 paper "No Free Lunch Theorems for Optimization". Wolpert had previously derived
No_free_lunch_theorem
Machine learning technique
then serves as a reward function to improve an agent's policy through an optimization algorithm like proximal policy optimization. RLHF has applications
Reinforcement learning from human feedback
Reinforcement_learning_from_human_feedback
Probabilistic optimization technique and metaheuristic
technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large
Simulated_annealing
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
Method to solve constrained optimization problems
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation
Lagrange_multiplier
Open source numerical analysis library
statistics, hypothesis testing) Multiple precision versions of linear algebra, interpolation and optimization algorithms (using MPFR for floating point computations)
ALGLIB
Competitive algorithm for searching a problem space
GA applications include optimizing decision trees for better performance, solving sudoku puzzles, hyperparameter optimization, and causal inference. In
Genetic_algorithm
Optimization technique for solving (mixed) integer linear programs
mathematical optimization, the cutting-plane method is any of a variety of optimization methods that iteratively refine a feasible set or objective function by
Cutting-plane_method
Quantum physics-based metaheuristic for optimization problems
Quantum annealing (QA) is an optimization process for finding the global minimum of a given objective function over a given set of candidate solutions
Quantum_annealing
Property of an intermediate representation in a compiler
variable may have received a value. Most optimizations can be adapted to preserve SSA form, so that one optimization can be performed after another with no
Static_single-assignment_form
Calculus on stochastic processes
stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This
Stochastic_calculus
Combinatorial optimization method
Branch and cut is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some
Branch_and_cut
Increasing execution speed and reducing the overheads associated with loops
representations of the computation being optimized and the optimization(s) being performed. Loop optimization can be viewed as the application of a sequence
Loop_optimization
Study of rates of change
to a differentiable function near a point. In this sense, differentiation is closely related to the differential. For functions of several variables
Differential_calculus
Microscopic optical resolution test device
A 1951 USAF resolution test chart is a microscopic optical resolution test device originally defined by the U.S. Air Force MIL-STD-150A standard of 1951
1951 USAF resolution test chart
1951_USAF_resolution_test_chart
Sequence of program instructions invokable by other software
as COBOL and BASIC, make a distinction between functions that return a value (typically called "functions") and those that do not (typically called "subprogram"
Function (computer programming)
Function_(computer_programming)
Overview of computer engineering topics
Power optimization (EDA) Timing closure Design flow (EDA) Design closure Rent's rule Design rule checking SystemVerilog In-circuit test Joint Test Action
Computer engineering compendium
Computer_engineering_compendium
Computer compiler optimization technique
Combinatorial Optimization, IPCO The Aussois Combinatorial Optimization Workshop Bosscher, Steven; and Novillo, Diego. GCC gets a new Optimizer Framework
Register_allocation
Mathematics of convex functions and sets
hyperplanes, and convex functions can be studied through supporting affine functions. Convex analysis is a common thread in modern optimization, duality theory
Convex_analysis
Scientific principles enabling the use of the calculus of variations
variations, which concerns finding functions that optimize the values of quantities that depend on those functions. For example, the problem of determining
Variational_principle
American mathematician, physicist and computer scientist
optimization methods and complex systems theory. One of Wolpert's most discussed achievements is known as No free lunch in search and optimization. By
David_Wolpert
Initial set of valid possible values
In mathematical optimization and computer science, a feasible region, feasible set, or solution space is the set of all possible points (sets of values
Feasible_region
Computer function
as the algorithms are optimized for their respective platforms. MurmurHash3 was released alongside SMHasher, a hash function test suite. The canonical
MurmurHash
Algorithm for linear programming
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is an algorithm for linear programming. The name of the algorithm is derived
Simplex_algorithm
publications are listed below for reference. One modern application of design optimization is structural design optimization (SDO) is in building and construction
Design_optimization
Machine learning system
with support for a number of machine learning reductions, importance weighting, and a selection of different loss functions and optimization algorithms
Vowpal_Wabbit
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
Field of machine learning
policy optimization (PPO), to produce outputs that the reward model scores highly. The reward model substitutes for human raters during optimization, so
Reinforcement_learning
Collection of C++ libraries
Functions". Proceedings of the 23rd ACM/IEEE International Symposium on Code Generation and Optimization. CGO '25. Las Vegas, Nevada: Association for
Abseil_(C++_libraries)
Concept in artificial intelligence
(RAG), develop specialized subsystems, or agents, each optimized for specific tasks and functions. Develop new and novel multimodal architectures that further
Recursive_self-improvement
Computer program for complexity reduction of digital logic circuits
number of output functions of a combinational function block. As a result, the Quine–McCluskey method is practical only for functions with a limited number
Espresso heuristic logic minimizer
Espresso_heuristic_logic_minimizer
Yes/no problem in computer science
in many cases the original function or optimization problem can be solved by solving its corresponding decision problem. For example, in the traveling
Decision_problem
Application software published by ETAS
vehicle, in the lab, on test benches or in combination with simulation environments, such as Simulink. A host of functions required for ECU software calibration
INCA_(software)
List of quantum computing algorithms
amplification, quantum walks, phase estimation, or hybrid quantum-classical optimization. Adiabatic quantum computation BQP Glossary of quantum computing List
List_of_quantum_algorithms
Feature of some compilers
user-defined functions in the same language. The Metacode extension to C++ (Vandevoorde 2003) was an early experimental system to allow compile-time function evaluation
Compile-time function execution
Compile-time_function_execution
Mathematical function, denoted exp(x) or e^x
distinguishing it from some other functions that are also commonly called exponential functions. These functions include the functions of the form f ( x ) = b
Exponential_function
Mapping arbitrary data to fixed-size values
functions, while cryptographic hash functions are used in cybersecurity to secure sensitive data such as passwords. In a hash table, a hash function takes
Hash_function
Information-theoretic measure
Other loss functions that penalize errors differently can be also used for training, resulting in models with different final test accuracy. For example
Cross-entropy
Recursive function
tak(z - 1, x, y) ) else: return z This function is often used as a benchmark for languages with optimization for recursion. The original definition by
Tak_(function)
Indicator function of positive numbers
be right-continuous. For instance cumulative distribution functions are usually taken to be right continuous, as are functions integrated against in
Heaviside_step_function
Python package
Problem #71 used to test the performance of nonlinear programming solvers. This particular optimization problem has an objective function min x ∈ R x 1 x
Gekko_(optimization_software)
relational optimizer implemented cost-based relational optimizer handle tables of 1,000 rows up to 10,000,000 rows user-defined functions (UDFs) within
IBM Peterlee Relational Test Vehicle
IBM_Peterlee_Relational_Test_Vehicle
On converting relations to functions of several real variables
the m variables yi are differentiable functions of the xj in some neighbourhood of the point. As these functions generally cannot be expressed in closed
Implicit_function_theorem
Sentence containing all letters of the English alphabet
English alphabet. Because of this, the phrase is commonly used for touch-typing practice, testing typewriters and computer keyboards, displaying examples of
The quick brown fox jumps over the lazy dog
The_quick_brown_fox_jumps_over_the_lazy_dog
Testing the qualities as opposed to the correctness of software
functional testing, which tests against functional requirements that describe the functions of a system and its components. Accessibility testing is a non-functional
Non-functional_testing
Set of methods for supervised statistical learning
loss and these other loss functions is best stated in terms of target functions - the function that minimizes expected risk for a given pair of random variables
Support_vector_machine
company in Scotts Valley, California, in the business of optimizing application performance for applications and embedded systems. Founded in 1993 by Chris
InterWorking_Labs
Subset of evolutionary computation
numerical optimization problems. Evolutionary multi-objective optimization – Applies evolutionary algorithms to multi-objective optimization problems,
Evolutionary_algorithm
being tested has an internal array or analog functions. Built-in self-test Built-in test equipment Design for test Power-on self-test Built-in Self Test (BIST)
Logic_built-in_self-test
Statistical distribution for dependence between random variables
portfolio management and optimization, and to derivatives pricing. For the former, copulas are used to perform stress-tests and robustness checks that
Copula_(statistics)
Statistical hypothesis test
Student's t-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It
Student's_t-test
Tasks in machine learning
on the training data set using a supervised learning method, for example using optimization methods such as gradient descent or stochastic gradient descent
Training, validation, and test data sets
Training,_validation,_and_test_data_sets
Algorithm in computer science
optimization technique. It uses the major genetic operators mutation, recombination and selection of parents. The 'evolution strategy' optimization technique
Evolution_strategy
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