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\((\Phi_ 1,\Phi_ 2)\)-convexity. (English) Zbl 0880.26009

Let \(F\) be a vector space of real-valued functions defined on a set \(D\subseteq \mathbb{R}^n\). A function \(f\in F\) is said to be \((\Phi_1,\Phi_2)\)-convex if \[ f(\Phi_1(x,y,\lambda))\leq \Phi_2(x,y,\lambda,f)\quad\text{for all }x,y\in D\quad\text{and }\lambda\in[0,1], \] where \(\Phi_1: D\times D\times [0,1]\to\mathbb{R}^n\) and \(\Phi_2:D\times D\times[0,1]\times F\to\mathbb{R}\) satisfy \[ \Phi_1(x,y,0)= y,\quad \Phi_1(x,x,\lambda)= x, \]
\[ \Phi_2(x,y,0,f)= f(y),\quad\Phi_2(x,x,\lambda,f)= f(x)\text{ and }\Phi_2(x,y,\lambda, f)\leq\max\{f(x), f(y)\}, \] and \(\Phi_1\) is assumed to be continuous in its third argument. One says that \(D\) is \(\Phi_1\)-convex if \(\Phi_1(x,y,\lambda)\in D\) for all \(x,y\in D\) and \(\lambda\in[0,1]\). The authors analyze optimization problems whose objective functions and constraint sets satisfy generalized convexity conditions of the above type; in particular, they obtain optimality conditions and study properties of perturbation functions. An associated notion of generalized monotonicity is also introduced; in the differentiable case, generalized convexity of functions is related to the generalized monotonicity of their gradients.

MSC:

26B25 Convexity of real functions of several variables, generalizations
90C30 Nonlinear programming
52A40 Inequalities and extremum problems involving convexity in convex geometry
49J52 Nonsmooth analysis
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References:

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