The global weak sharp minima with explicit exponents in polynomial vector optimization problems (Q1746026)

The authors consider the following vector optimization problems: \[ \begin{aligned} V\text{-minimize }f(x):&=(f_1(x),\dots,f_m(x)),\\ \text{subject to }g_i(x)&\leq 0, \quad i=1,\dots,r, \\ h_j(x)&=0,\quad j=1,\dots,s, \end{aligned}\eqno{(1)} \] where \(f_k,g_i,h_j:\mathbb R^n\to\mathbb R\) are polynomials of degree at most \(d\). Let \(\Omega\) be the feasible set of (1) and set \[ S_w=\{x\in\Omega:(f(\Omega)-f(x))\cap(-\mathrm{int}\mathbb R_+^m)=\emptyset\}. \] Define the function \(\psi:\mathbb R^n\to\mathbb R\) by \[ \Psi(x)=\sup_{y\in\Omega}\min_{k=1,\dots,m}(f_k(x)-f(y)). \] The authors prove that this function is nonnegative on \(\Omega\) and also \(S_w\) is the kernel of \(\psi\) on \(\Omega\), i.e., \(S_w=\{x\in\Omega:\psi(x)=0\}\). By considering the problem (1), the authors generalize the classical Lojasiewicz inequality to the global weak sharp minima with explicit exponents as follows: Suppose that the Mangasarian-Fromovitz constraint qualification holds on \(\Omega\) and that the restriction of \(f\) on \(\Omega\) is ``regular at infinity''. Then there exist \(c>0\) and \(\alpha>0\) such that \[ \mathrm{c} \; \mathrm{dist}(x,S_w)\leq[\psi(x)]^{\alpha}+\psi(x)\quad \forall x\in\Omega, \] where \(\mathrm{dist}(x,S_w)\) denotes the distance from \(x\) to \(S_w\) and \(\alpha=(1/(R((n+m+r+s+2)(n+1)+r+s-1,d+4)))\). Note that \(R:\mathbb N\times\mathbb N\to\mathbb N\) is defined by \(R(p,q)=\{q(3q-3)^{p-1} \; \mathrm{if} \; q>2 \; \mathrm{or} \; 1 \; \mathrm{if} \; q=1\}\). In the last section of this paper, some examples are offered to illustrate the main result.