Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers
The authors provide a new generalized Lagrange duality concept for (not necessary convex) optimization problems where the objective function and the constraint functions are polynomials and where also the Lagrange multipliers are replaced by polynomials. In detail, let \(\mathbb{R}[x]\) be the ring of real polynomials in \(x\in\mathbb{R}^n\) and \(\sum_n\) be the set of functions which are the sum of square polynomials, i.e., functions \(f=\sum^r_{j=1}(f_j)^2\) with \(f_j\in\mathbb{R}[x]\). Starting with the primal optimization problem \[f(x)\to\min\quad\text{s.t.}\quad h_i(x)= 0\,\, (i= 1,\dots,p),\quad g_j(x)\ge 0\,\, (j=1,\dots,m)\] (where all functions \(f\), \(h_i\), \(g_j\in\mathbb{R}[x]\)), the associated polynomial Lagrangian function is introduced by \[L(x,\phi,\sigma):=f(x)- \sum^p_{i=1} \phi_i(x) h)i(x)- \sum^m_{j=1} \sigma_j(x) g_j(x)\] and the dual optimization problem by \[\inf\{L(x,\phi,\sigma\mid x\in\mathbb{R}^n\}\to\max\quad\text{s.t. }\phi_i\in \mathbb{R}[x]\ (i= 1,\dots,p),\ \sigma_j\in\sum_n\ (j= 1,\dots,m).\] Using a suitable nonnegativity certificate, the authors provide necessary and sufficient optimality conditions, but also weak and strong duality assertions in connection with the presence of saddle points of the introduced Lagrangian. Several numerical examples illustrate the results.
scientific article; zbMATH DE number 6996809
scientific article