Karush–Kuhn–Tucker conditions

In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints. Similar to the Lagrange approach, the constrained maximization (minimization) problem is rewritten as a Lagrange function whose optimal point is a (global) saddle point, i.e. a global maximum (minimum) over the domain of the choice variables and a global minimum (maximum) over the multipliers, which is why the Karush–Kuhn–Tucker theorem is sometimes referred to as the saddle-point theorem. The KKT conditions were originally named after Harold W. Kuhn and Albert W. Tucker, who first published the conditions in 1951. Later scho

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (39)

Related publications

Related people (4)

Related people

Related units (3)

Related units

Related concepts (10)

Related concepts

Related courses (35)

Related courses

Related lectures (134)

Related lectures

Mathematical optimization

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternative

Duality (optimization)

In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem

Lagrange multiplier

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition th

High-dimensional optimization under nonconvex excluded volume constraints

Antonio Sclocchi

We consider high-dimensional random optimization problems where the dynamical variables are subjected to nonconvex excluded volume constraints. We focus on the case in which the cost function is a simple quadratic cost and the excluded volume constraints are modeled by a perceptron constraint satisfaction problem. We show that depending on the density of constraints, one can have different situations. If the number of constraints is small, one typically has a phase where the ground state of the cost function is unique and sits on the boundary of the island of configurations allowed by the constraints. In this case, there is a hypostatic number of marginally satisfied constraints. If the number of constraints is increased one enters a glassy phase where the cost function has many local minima sitting again on the boundary of the regions of allowed configurations. At the phase transition point, the total number of marginally satisfied constraints becomes equal to the number of degrees of freedom in the problem and therefore we say that these minima are isostatic. We conjecture that by increasing further the constraints the system stays isostatic up to the point where the volume of available phase space shrinks to zero. We derive our results using the replica method and we also analyze a dynamical algorithm, the Karush-Kuhn-Tucker algorithm, through dynamical mean-field theory and we show how to recover the results of the replica approach in the replica symmetric phase.

AMER PHYSICAL SOC2022

Central limit theorems for multilevel Monte Carlo methods

Hakon Andreas Hoel, Sebastian Krumscheid

In this work, we show that uniform integrability is not a necessary condition for central limit theorems (CLT) to hold for normalized multilevel Monte Carlo (MLMC) estimators and we provide near optimal weaker conditions under which the CLT is achieved. In particular, if the variance decay rate dominates the computational cost rate (i.e., beta > gamma), we prove that the CLT applies to the standard (variance minimizing) MLMC estimator. For other settings where the CLT may not apply to the standard MLMC estimator, we propose an alternative estimator, called the mass-shifted MLMC estimator, to which the CLT always applies. This comes at a small efficiency loss: the computational cost of achieving mean square approximation error O(epsilon(2)) is at worst a factor O(log(1/epsilon)) higher with the mass-shifted estimator than with the standard one. (C) 2019 Elsevier Inc. All rights reserved.

2019

Central limit theorems for multilevel Monte Carlo methods

Hakon Andreas Hoel, Sebastian Krumscheid

In this work, we show that uniform integrability is not a necessary condition for central limit theorems (CLT) to hold for normalized multilevel Monte Carlo MLMC estimators, and we provide near optimal weaker conditions under which the CLT is achieved. In particular, if the variance decay rate dominates the computational cost rate (i.e.,

\beta> \gamma

), we prove that the CLT always holds.

2018

MATH-329: Continuous optimization

This course introduces students to continuous, nonlinear optimization. We study the theory of optimization with continuous variables (with full proofs), and we analyze and implement important algorithms to solve constrained and unconstrained problems.

MATH-351: Advanced numerical analysis

The student will learn state-of-the-art algorithms for solving differential equations. The analysis and implementation of these algorithms will be discussed in some detail.

MGT-483: Optimal decision making

This course introduces the theory and applications of optimization. We develop tools and concepts of optimization and decision analysis that enable managers in manufacturing, service operations, marketing, transportation and finance to transform data into insights for making better decisions.