\((\Phi_ 1,\Phi_ 2)\)-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.


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
Full Text: DOI


[1] DOI: 10.1007/BF00939383 · Zbl 0802.49027
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