Slater's condition

Let $f\_1,\backslash ldots,f\_m$ be real-valued functions on some subset $D$ of $\backslash mathbb\{R\}^n$. We say that the functions satisfy the Slater condition if there exists some $x$ in the relative interior of $D$, for which for all $i$ in $1,\backslash ldots,m$. We say that the functions satisfy the relaxed Slater condition if:{{Cite web |last=Nemirovsky and Ben-Tal |date=2023 |title=Optimization III: Convex Optimization |url=http://www2.isye.gatech.edu/~nemirovs/OPTIIILN2023Spring.pdf}}

• Some $k$ functions (say $f\_1,\backslash ldots,f\_k$) are affine;
• There exists $x\; \backslash in\; \backslash operatorname\{relint\}\; D$ such that $f\_i(x)\; \backslash le\; 0$ for all $i=1,\backslash ldots,k$, and for all $i=k+1,\backslash ldots,m$.

Consider the optimization problem

:$\backslash text\{Minimize\; \}\backslash ;\; f\_0(x)$

:$\backslash text\{subject\; to:\; \}\backslash$

::$f\_i(x)\; \backslash le\; 0\; ,\; i\; =\; 1,\backslash ldots,m$

::$Ax\; =\; b$

where $f\_0,\backslash ldots,f\_m$ are convex functions. This is an instance of convex programming. Slater's condition for convex programming states that there exists an $x^*$ that is strictly feasible, that is, all m constraints are satisfied, and the nonlinear constraints are satisfied with strict inequalities.

If a convex program satisfies Slater's condition (or relaxed condition), and it is bounded from below, then strong duality holds. Mathematically, this states that strong duality holds if there exists an $x^*\; \backslash in\; \backslash operatorname\{relint\}(D)$ (where relint denotes the relative interior of the convex set

$D\; :=\; \backslash cap\_\{i\; =\; 0\}^m\; \backslash operatorname\{dom\}(f\_i)$) such that

: (the convex, nonlinear constraints)

:$Ax^*\; =\; b.\backslash ,${{cite book |last1=Boyd |first1=Stephen |last2=Vandenberghe |first2=Lieven |title=Convex Optimization |publisher=Cambridge University Press |year=2004 |isbn=978-0-521-83378-3 |url=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=240 |format=pdf |accessdate=October 3, 2011}}

Given the problem

:$\backslash text\{Minimize\; \}\backslash ;\; f\_0(x)$

:$\backslash text\{subject\; to:\; \}\backslash$

::$f\_i(x)\; \backslash le\_\{K\_i\}\; 0\; ,\; i\; =\; 1,\backslash ldots,m$

::$Ax\; =\; b$

where $f\_0$ is convex and $f\_i$ is $K\_i$-convex for each $i$. Then Slater's condition says that if there exists an $x^*\; \backslash in\; \backslash operatorname\{relint\}(D)$ such that

: and

:$Ax^*\; =\; b$

then strong duality holds.

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• Duality
• Karush–Kuhn–Tucker conditions
• Lagrange multiplier

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Category:Convex optimization