Minimizing Convex Functions with Bounded Perturbations

• Authors: Hoang Xuan Phu
• Venue: SIAM Journal on Optimization
• Year: 2010

We investigate the problem of minimizing the perturbed convex function $\tilde{f}(x)=f(x)+p(x)$ over some convex subset $D$ of a normed linear space $X$, where the function $f$ is convex and the perturbation $p$ is bounded. The key tool for our investigation is a convexity modulus of $f$ named $h_1$, whose generalized inverse function $h_1^{-1}$ is used to define the quantity $\gamma^*:=h_1^{-1}(2\sup_{x\in D}|p(x)|)$. Generally, by the irregular perturbation $p$, the perturbed function $\tilde{f}$ loses all usual analytical and optimization properties yielded by the convexity of $f$. But we show that some convexity trace remains in $\tilde{f}$, namely, $\tilde{f}$ is outer $\gamma$-convex for any $\gamma\geq\gamma^*$ and strictly outer $\gamma$-convex for any $\gamma\gamma^*$. As a consequence, each $\gamma^*$-minimizer $x^*\in D$ defined by $\tilde{f}(x^*)=\inf_{x\in\bar{B}(x^*,\gamma^*)\cap D}\tilde{f}(x)$ is a global minimizer, i.e., $\tilde{f}(x^*)=\inf_{x\in D}\tilde{f}(x)$, and each $\gamma^*$-infimizer $x^*$ defined by ${\lim\inf}_{x\in D,\,x\to x^*}\tilde{f}(x)=\inf_{x\in\bar{B}(x^*,\gamma^*)\cap D}\tilde{f}(x)$ is a global infimizer, i.e., ${\lim\inf}_{x\in D,\,x\to x^*}\tilde{f}(x)=\inf_{x\in D}\tilde{f}(x)$. Moreover, the diameter of the set of global infimizers (including global minimizers) of $\tilde{f}$ is not greater than $\gamma^*$, and the distance between any global infimizer of $\tilde{f}$ and any global infimizer of $f$ cannot exceed $\gamma^*$. The latter property is used for sensibility analysis.