= Solution
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 $\|\nabla f_{\mu_k}(x^k)\|\leq\varepsilon_k$, where $\varepsilon_k\to0$. Set $\mu_{k+1}=\theta\mu_k$ and use $x^k$ as the next starting point.
The associated <Lagrange multiplier> estimates are
$$
\lambda_i^k=\frac{\mu_k}{c_i(x^k)}>0,\qquad\lambda_i^kc_i(x^k)=\mu_k.
$$
Reducing $\mu_k$ 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.
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