A barrier function method for minimax problems (Q1186276)

The authors consider the optimization problem to minimize \(\psi(x)\) on \(\mathbb{R}^ n\) where \[ \psi(x):=\max\left(f^ 1(x),\dots,f^ m(x), \max_{t\in[0,1]}\Phi^ 1(x,t),\dots,\max_{t\in[0,1]}\Phi^ \ell(x,t)\right) \] and \(f^ j\), \(j=1,\dots,m\), \(\Phi^ k\), \(k=1,\dots,\ell\), are continuously differentiable functions. Such optimization problems occurs in engineering design problems, where \(\Phi^ j(x,t)\) arises from constraints on time or frequency response. The presented algorithm is based on the barrier function \[ p(\alpha,x)=\sum^ m_{j=1}{1\over (\alpha-f^ j(x))}+\sum^ \ell_{k=1}\int_{[0,1]}{1\over (\alpha-\Phi^ k(x,t))} dt, \] where \(\alpha>\psi(x)\). In the \(i\)-th iteration the algorithm takes \(\alpha_ i=\psi(x_ i)\) and computes \(x_{i+1}\in\arg\min_{x\in C(\alpha_ i)} p(\alpha_ i,x)\), where \(C(\alpha_ i)=\{x\in\mathbb{R}^ n\mid\psi(x)<\alpha_ i\}\). In an ``implementable version'', the algorithm uses the points \(x_ i\) and \(x_{i-1}\). Any accumulation point \(\hat x\) of the generated sequence \((x_ i)\) satisfies \(0\in\partial\psi(\hat x)\), where \(\partial\psi(\hat x)\) denotes the generalized gradient of Clarke. The algorithm does not need any special search direction routine, has a simple structure, and requires small memory. Test examples from the literature illustrate that the algorithm converges linearly, that its computing times are comparable to those of other algorithms, but does not fail when others do.