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A barrier function penalizes approaching the boundary of a feasible set from its interior, typically diverging to $+\infty$ there. An interior-point method minimizes the original objective function plus a positive multiple of this penalty, then reduces that multiple. The logarithmic barrier is the basic example for smooth inequality optimization constraints.

Let $c_i$ be concave and continuously differentiable, and let a strict feasible point $z$ satisfy $\delta ={\min }_i{c}_i(z)>0$. Suppose $x_k\to x_{\ast }$, $c_i(x_k)>0$, ${\mu }_k\to 0$, and the logarithmic barrier residuals $r_k=\nabla f(x_k)-{\sum }_i{\lambda }_{k,i}\nabla c_i(x_k)$ tend to zero, where ${\lambda }_{k,i}={\mu }_k/c_i(x_k)$. By concavity,The right-hand side is bounded when $\nabla f$ is continuous near $x_{\ast }$. Thus the nonnegative Lagrange multiplier estimates are bounded. A convergent subsequence gives ${\lambda }_{\ast }\ge 0$; passing to the limit in stationarity and ${\lambda }_{k,i}{c}_i(x_k)={\mu }_k$ proves the Karush-Kuhn-Tucker conditions at $x_{\ast }$. An inactive optimization constraint has $c_i(x_{\ast })>0$ and hence ${\lambda }_{k,i}\to 0$ even without selecting a subsequence. With a convex objective function, these conditions certify a global minimum. This result is conditional on the stated residual and primal convergence assumptions; it does not assert that every barrier subproblem has a minimizer.

For optimization constraints $c_i(x)\ge 0$ and $\mu >0$, the logarithmic barrier objective is $f_{\mu }(x)=f(x)-\mu {\sum }_i\log c_i(x)$ on $D=\{x:c_i(x)>0\forall i\}$. Its derivatives are defined only on this strictly feasible open subset. One may extend its value to $+\infty$ off $D$ to formulate an extended-real convex function. The penalty alone is $-{\sum }_i\log c_i$, while $f_{\mu }$ includes the original objective function.

When $f$ is convex and each $c_i$ is concave, $D$ is a convex set andEvery term of this Hessian matrix is a positive semidefinite matrix, so $f_{\mu }$ is convex. Its critical point satisfies the stationarity part of the Karush-Kuhn-Tucker conditions with estimates ${\lambda }_i=\mu /c_i(x)>0$ and perturbed complementary slackness ${\lambda }_i{c}_i(x)=\mu$.

For $\mu >0$, the logarithmic barrier objective isThe domain of a function $D$ is a nonempty convex set by the same concavity argument as in part (a), with strict inequalities. Differentiating twice on $D$ givesHere ${\nabla }^{2}f⪰0$ by convexity, $-{\nabla }^2{c}_i⪰0$ by concavity, and each outer product is a positive semidefinite matrix. Hence ${\nabla }^2{f}_{\mu }⪰0$, proving convexity on $D$.

The domain qualification matters: the displayed real-valued logarithm expression is not defined on all of ${\mathbb{R}}^n$. The precise whole-space version is the extended-real convex function equal to $f_{\mu }$ on $D$ and $+\infty$ outside $D$.

Choose a strictly feasible point, an initial barrier parameter ${\mu }_0>0$, and $0<\theta <1$. At outer iteration $k$, approximately minimize the logarithmic barrier objective $f_{{\mu }_k}$ over its strictly feasible domain of a function. For example, use damped Newton methods, choosing an inner step size by backtracking line search so that every $c_i$ stays positive and the barrier objective function decreases. Solve to a tolerance $\Vert \nabla f_{{\mu }_k}(x^k)\Vert \le {\epsilon }_k$, where ${\epsilon }_k\to 0$. Set ${\mu }_{k+1}=\theta {\mu }_k$ and use $x^k$ as the next starting point.

The associated Lagrange multiplier estimates areReducing ${\mu }_k$ makes these perturbed complementary slackness equations approach the KKT conditions. This describes the generic interior-point method; existence of a solution to every inner minimization needs appropriate additional attainment assumptions and does not follow from strict feasibility alone.