Questions tagged [karush-kuhn-tucker]

In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions are first order necessary conditions for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied.

In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are 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. The system of equations and inequalities corresponding to the KKT conditions is usually not solved directly, except in the few special cases where a closed-form solution can be derived analytically. In general, many optimization algorithms can be interpreted as methods for numerically solving the KKT system of equations and inequalities.

The KKT conditions were originally named after Harold W. Kuhn, and Albert W. Tucker, who first published the conditions in 1951. Later scholars discovered that the necessary conditions for this problem had been stated by William Karush in his master's thesis in 1939.

The KKT conditions include stationarity, primal feasibility, dual feasibility, and complementary slackness.

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When do constraint qualifications imply strong duality?

Assumptions / Conventions I am interested in problems of the form $$ \inf_{x \in \mathbb{R}^n} ~ f_0(x) ~:~ f_i(x) \leq 0 ~\forall ~i \in [m], ~~ g_i(x) = 0 ~\forall ~i \in [k]$$ where $f_i$ are convex and differentiable, and $g_i$ are affine. I…
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How to use Karush-Kuhn-Tucker (KKT) conditions in inequality constrained optimization

I am trying to understand how to use the Karush-Kuhn-Tucker conditions, similar as asked but not answered in this thread. Assume the target function is given by $f(x)$, where $x$ a vector. Let $g(x) \ge 0 $ be an inequality constraint under which…
tomka
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How to use the Karush–Kuhn–Tucker conditions?

From what I read, the Karush-Kuhn-Tucker conditions are a generalization of the Lagrange Multiplier Method. For the Lagrange Multiplier Method I have been able to find a serie of steps I must do to find the result, but I don't see quite clearly…
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Non-trivial example of a local optimum without satisfying Karush-Kuhn-Tucker (KKT) optimality conditions

I am looking for an example of a locally optimal point of the nonlinear program: $$\{ \min f(x) , g_i(x) \geq 0 \}$$ that does not have a singleton feasibility set. e.g. not the example of $\{ \min x, x^2\leq 0 \} $ found here where the feasibility…
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I have a problem from Karush-Kuhn-Tucker condition

maximize: $xy-x$ and s.t. $x^2+y^2 \leq 9$ and $x \ge 0$ Then, I tried to find points using lagrange method. I used 4 cases and they are $(1)$ $\lambda_1 =0$ and $\lambda_2 = 0$ forrect?
Jumbo09
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KKT-Conditions in a functional setting

Let $F:L^2([0,1])\rightarrow \mathbb{R}$ be a convex functional. Consider the minimization problem \begin{align} \underset{f(\cdot) \in L^2([0,1])}{\min} F(f)\,\,\text{ subject to } \|f(\cdot) \|_2\leq \lambda, \end{align} with $\lambda>0$…
RobH
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How to project a vector on a set defined by linear inequality constraints through KKT conditions?

I need to find the projection $x \in \mathbb{R}^{k}$ of a vector $z \in \mathbb{R}^{k}$ on the set defined by $Y \cdot x \geq 0$ where $Y$ is a (given but no specific property) matrix of size $m \cdot k$. I first build the Lagrangian as follows…
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Reference Request: KKT in Hilbert Space

Are there analogues of Slater's condition and the KKT conditions in separable Hilbert spaces? Does the infinite dimensionality pose a problem?
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Confusion about KKT points

I'm confused about sufficiency of KKT conditions for optimality. In http://www.stat.cmu.edu/~ryantibs/convexopt/scribes/kkt-scribed.pdf, page 3, Section 12.1.3 it is stated that For any optimization problem, if $x^\star$ and $(u^\star,v^\star)$ …
pulosky
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Geometric Meaning of KKT Conditions

I am trying to state KKT conditions for an optimization with both inequality and equality constraints in geometric form, so that I get the big picture better. Is this what it says? At the optimal solution, the gradient of the function must be a…
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Is this a known result?

I heard the following result and I am wondering if anyone can verify its correctness and also provide a source to cite. If the Lagrangian $L(x,\lambda)$ is convex in $x$ at the optimal Lagrange multiplier $\lambda^*$, that is, $L(x,\lambda^*)$ is…
jonem
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Is there a nice representation for KKT conditions for matrix constraints?

I have a convex programming problem: $\min \left\lVert J - R \right\rVert _F$ $J,R$ are matrices. $J$ is given for the problem. One of the constraints is: $R = KQ$ Here, $R,K,Q$ are matrices. $K$ is given. $Q$ is a variable along with $R$. I…
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Optimization Problem: Karush-Kuhn-Tucker Condition

I am working on the question displayed below. I know that the method of Lagrange Multipliers is used to find the solution for optimization problems constrained to one or more equalities and when our constraints also have inequalities, we need to…
Shady
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KKT conditions for $\max \log \det(X)$ with LMI constraints

I am trying to derive the KKT conditions for the following convex optimization problem where $A$ is a given matrix: $$\begin{array}{ll} \underset{X,Y,Z}{\text{minimize}} & - \log \det \left(I + Z + A X A^T + Y A^T + A Y^T \right)\\ \text{subject to}…
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Karush Kuhn Tucker and Optimal Minimum

I am a little not clear on the solutions of KKT Conditions. Suppose we have a convex function $f(x)$ and at a specific $x$ are our KKT conditions fulfilled. Does this make this point a global minimum of our function, or just a local minimum that…
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