## $$(\Phi_ 1,\Phi_ 2)$$-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
Full Text:

### References:

 [1] DOI: 10.1007/BF00939383 · Zbl 0802.49027 [2] DOI: 10.1007/BF00939919 · Zbl 0793.90069 [3] DOI: 10.1007/BF00939383 · Zbl 0802.49027 [4] DOI: 10.1007/BF00932539 · Zbl 0325.26007 [5] DOI: 10.1007/BF02592078 · Zbl 0643.90071 [6] DOI: 10.1006/jmaa.1995.1057 · Zbl 0831.90097 [7] DOI: 10.1016/0022-247X(88)90135-7 · Zbl 0667.49001 [8] DOI: 10.1080/02331939408843959 · Zbl 0816.26004 [9] Rapcsak T., J .O. T. A 69 pp 69– (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.