Adjoint coexhausters in nonsmooth analysis and extremality conditions.

*(English)*Zbl 1280.90112The article of M. E. Abbasov and V. F. Demyanov is a valuable contribution to the field of nondifferentiable or nonsmooth calculus and, especially, optimization. This investigation takes place in a wide analytic setting, in the tradition of the quasidifferential which it extends and employs. This work is well structured, well written and, e.g., by figures, well visualized.

Classically, in “smooth” mathematical calculus and optimization, a differentiable function is studied by means of the derivative or, multidimensionally, the gradient. In the case of nondifferentiable functions, nonsmooth analysis gets used. In convex analysis and minimax theory, the corresponding classes of functions are investigated by the subdifferential, quasidifferentiable functions are evaluated by the quasidifferential (being is a pair of sets). Any directionally differentiable function is treated in terms of upper and lower exhausters (families of convex sets). Here, conditions for a minimizer are given with an upper exhauster; while conditions for a maximizer are represented by a lower exhauster. Hence, upper exhausters are called proper ones for the minimization whereas a lower exhauster is referred to as a proper one for the maximization task. The directional derivatives and so, the exhausters permit first-order approximations of the increment of the regarded function. As functions of the direction, these approximations are positively homogeneous. They permit the stating of optimality conditions, the finding of steepest ascent and descent directions and the constructing of numerical procedures. If, however, e.g., the maximizer of the function has to be found, but one has an upper exhauster (being not proper for the maximization), it is required to employ a lower exhauster. One could try to express conditions for a maximum in expressions of an upper exhauster (which is an adjoint one for the maximization). First research on this was done by Roshchina, and then Abbasov. Generally, exhauster mappings are discontinuous (in Hausdorff sense), so that computational problems can occur. To overcome these obstacles, upper and lower coexhausters are introduced, allowing first-order approximations of the increment of the function; now, they do not need to be positively homogeneous. Those approximations permit the statement of optimality conditions, the finding of ascent and descent directions (not the steepest ones, however), the design of numerical methods with good properties of convergence. Conditions for a minimum are given by an upper coexhauster (called a proper coexhauster for the minimization), whereas conditions for a maximum are presented with a lower coexhauster (called a proper one for the maximization). The authors derive optimality conditions in expressions of adjoint coexhausters.

The six sections of this article are as follows: 1. Introduction, 2. Directional differentiability, 3. Exhausters, 4. Coexhausters, 5. Optimality conditions in terms of adjoint coexhausters, and 6. Concluding remarks.

In the future, further strong results and numerical techniques could be expected, initialized the present research paper. Those advances might foster and initiate emerging achievements in science, e.g. in statistics and data mining, in engineering, economics, decision making, finance and OR, in medicine and healthcare, and, eventually, to improvements of living conditions of the peoples.

Classically, in “smooth” mathematical calculus and optimization, a differentiable function is studied by means of the derivative or, multidimensionally, the gradient. In the case of nondifferentiable functions, nonsmooth analysis gets used. In convex analysis and minimax theory, the corresponding classes of functions are investigated by the subdifferential, quasidifferentiable functions are evaluated by the quasidifferential (being is a pair of sets). Any directionally differentiable function is treated in terms of upper and lower exhausters (families of convex sets). Here, conditions for a minimizer are given with an upper exhauster; while conditions for a maximizer are represented by a lower exhauster. Hence, upper exhausters are called proper ones for the minimization whereas a lower exhauster is referred to as a proper one for the maximization task. The directional derivatives and so, the exhausters permit first-order approximations of the increment of the regarded function. As functions of the direction, these approximations are positively homogeneous. They permit the stating of optimality conditions, the finding of steepest ascent and descent directions and the constructing of numerical procedures. If, however, e.g., the maximizer of the function has to be found, but one has an upper exhauster (being not proper for the maximization), it is required to employ a lower exhauster. One could try to express conditions for a maximum in expressions of an upper exhauster (which is an adjoint one for the maximization). First research on this was done by Roshchina, and then Abbasov. Generally, exhauster mappings are discontinuous (in Hausdorff sense), so that computational problems can occur. To overcome these obstacles, upper and lower coexhausters are introduced, allowing first-order approximations of the increment of the function; now, they do not need to be positively homogeneous. Those approximations permit the statement of optimality conditions, the finding of ascent and descent directions (not the steepest ones, however), the design of numerical methods with good properties of convergence. Conditions for a minimum are given by an upper coexhauster (called a proper coexhauster for the minimization), whereas conditions for a maximum are presented with a lower coexhauster (called a proper one for the maximization). The authors derive optimality conditions in expressions of adjoint coexhausters.

The six sections of this article are as follows: 1. Introduction, 2. Directional differentiability, 3. Exhausters, 4. Coexhausters, 5. Optimality conditions in terms of adjoint coexhausters, and 6. Concluding remarks.

In the future, further strong results and numerical techniques could be expected, initialized the present research paper. Those advances might foster and initiate emerging achievements in science, e.g. in statistics and data mining, in engineering, economics, decision making, finance and OR, in medicine and healthcare, and, eventually, to improvements of living conditions of the peoples.

##### MSC:

90C30 | Nonlinear programming |

90C46 | Optimality conditions and duality in mathematical programming |

##### Keywords:

nonsmooth analysis; directional derivatives; proper and adjoint exhausters and coexhausters; extremality conditions
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\textit{M. E. Abbasov} and \textit{V. F. Demyanov}, J. Optim. Theory Appl. 156, No. 3, 535--553 (2013; Zbl 1280.90112)

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##### References:

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