 Solution

A precise conditional convergence statement is: with the smoothness and convexity assumptions of the question, a strict feasible point, ${\mu }_k\downarrow 0$, and barrier stationarity residualsevery convergent sequence of strictly feasible points $x^k\to x_{\ast }$ has a global minimizer as its limit. The Lagrange multiplier estimates are bounded and have a subsequence converging to multipliers satisfying the KKT conditions. The theorem does not assert convergence of the primal sequence without further assumptions.

First, continuity gives $c_i(x_{\ast })\ge 0$. For an inactive optimization constraint, $c_i(x_{\ast })>0$, soFor the active constraints, the hint permits assuming ${\lambda }_i^k\to {\lambda }_i^{\ast }$. Combining this with the inactive limits gives ${\lambda }_{\ast }\ge 0$, and passage to the limit in the residual equation proves stationarity. Since ${\lambda }_i^k{c}_i(x^k)={\mu }_k\to 0$, complementary slackness also holds. All KKT conditions follow, and part (b) proves global optimality.

The hint's multiplier assumption can in fact be weakened to selecting a subsequence. To see why the estimates are bounded, choose a strict feasible point $z$ and put $\delta ={\min }_i{c}_i(z)>0$. Concavity givesThe right side is bounded because $x^k\to x_{\ast }$ and $r_k\to 0$. This is the bounded barrier multipliers under Slater's condition argument. A convergent multiplier subsequence then gives the same KKT point, which suffices to prove that the assumed primal limit is globally optimal.