arXiv subjects

Majid E. Abbasov

Publications and source records attributed to Majid E. Abbasov.

6 recordsLinked to original sources

arXiv2026

The Unified Separation Condition: A General Constraint Qualification for Smooth and Nonsmooth Optimization

We introduce the Unified Separation Condition (USC) - a simple and general constraint qualification for finite-dimensional nonlinear programming. The USC requires that the origin be uniformly separated from the subdifferential of the constraint violation function in a neighborhood of the feasible set. The main result of this paper is two-fold. First, we show that the classical Mangasarian--Fromovitz constraint qualification (MFCQ) for inequalities and the linear independence constraint qualification (LICQ) for equalities imply the USC. Conversely, USC is strictly more general: it remains applicable in nonsmooth settings where classical conditions are not defined, and it may hold even when MFCQ fails. The proof that MFCQ implies USC is based on Gordan's theorem; the implication from LICQ to USC follows from the linear independence of the gradients of the active constraints. We also introduce a local version of USC and discuss its computational verification via convex quadratic programming. The USC provides a unified framework for constraint qualifications, bridging classical smooth theory and nonsmooth optimization.

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arXiv2026

Optimal 3D Road Alignment on Topographic Surfaces: A Convergent Dynamic Programming Approach

We consider the problem of finding an optimal 3D road trajectory between two points on a terrain with variable elevation. Unlike common heuristic pathfinding methods, we propose a rigorous framework based on the calculus of variations, introducing an integral cost functional that incorporates material delivery and construction expenses. The existence of a global minimizer is established via the Arzelà--Ascoli theorem. To solve the problem numerically, we develop a dynamic programming scheme and provide a formal convergence proof. We prove that the sequence of piecewise-linear solutions converges to the true optimum when the grid discretization steps follow a specific power-law relation -- specifically, when the vertical step size decays faster than the horizontal one. To enhance efficiency, we introduce a local-search modification that reduces computational complexity to nearly quadratic O(τ^{-2-ε}), where τ is the discretization step along the x-axis. Numerical experiments on 2D and 3D terrains validate the theoretical results, showing that our approach achieves accuracy comparable to the Ritz method while significantly reducing processing time.

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arXiv2025

Converting exhausters and coexhausters

Exhausters and coexhausters are notions of constructive nonsmooth analysis which are used to study extremal properties of functions. An upper exhauster (coexhauster) is used to get an approximation of a considered function in the neighborhood of a point in the form of $\min\max$ of linear (affine) functions. A lower exhauster (coexhauster) is used to represent the approximation in the form of $\max\min$ of linear (affine) functions. Conditions for a minimum in a most simple way are expressed by means of upper exhausters and coexhausters, while conditions for a maximum are described in terms of lower exhausters and coexhausters. Thus the problem of obtaining an upper exhauster or coexhauster when the lower one is given and vice verse arises. We study this problem in the paper and propose new method for its solution which allows one to pass easily between $\min\max$ and $\max\min$ representations.

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arXiv2025

Regularization for the Approximation of 2D Set of Points via the Length of the Curve

We study the problem of approximation of 2D set of points. Such type of problems always occur in physical experiments, econometrics, data analysis and other areas. The often problems of outliers or spikes usually make researchers to apply regularization techniques, such as Lasso, Ridge or Elastic Net. These approaches always employ penalty coefficient. So the important question of evaluation of the upper bound for the coefficient arises. In the current study we propose a novel way of regularization and derive the upper bound for the used penalty coefficient. First the problem in a general form is stated. The solution is sought in the class of piecewise continuously differentiable functions. It is shown that the optimal solution belongs to the class of piecewise linear functions. So the problem of obtaining the piecewise linear approximation that fits 2D set of point the best is stated. We show that the optimal solution is trivial and tends to a line as penalty coefficient tends to infinity. Then the main result is stated and proved. It provides the upper bound for the penalty coefficient prior to which the optimal solution differs from the line more than some pregiven positive number. We also demonstrate the proposed ideas on numerical examples which include comparison with other regularization approaches.

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arXiv2021

Directional differentiability, coexhausters, codifferentials and polyhedral DC functions

Codifferentials and coexhausters are used to describe nonhomogeneous approximations of a nonsmooth function. Despite the fact that coexhausters are modern generalizations of codifferentials, the theories of these two concepts continue to develop simultaneously. Moreover, codifferentials and coexhausters are strongly connected with DC functions. In this paper we trace analogies between all these objects, and prove the equivalence of the boundedness and optimality conditions described in terms of these notions. This allows one to extend the results derived in terms of one object to the problems stated via the other one. Another contribution of this paper is the study of connection between nonhomogeneous approximations and directional derivatives and formulate optimality conditions in terms of nonhomogeneous approximations.

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arXiv2020

Finding the set of global minimizers of a piecewise affine function

In the present work we study a problem of finding a global minimum of a piecewise affine function. We employ optimality conditions for the problem in terms of coexhausters and use them to state and prove necessary and sufficient conditions for a piecewise affine function to be bounded from below. We construct a simple method based on these conditions which allows one to get the minimum value of a studied function and the corresponding set of all its global minimizers. These results are built via coexhauster notion. This notion was introduced by V. F. Demyanov. Coexhausters are families of convex compact sets that allow one to represent the approximation of the increment of the studied function at a considered point in the form of minmax or maxmin of affine functions. We take these representations as a definition of a piecewise affine function and show that they correspond with the definitions for piecewise affine function given by other researchers. All the conditions and methods were obtained by means of coexhausters theory. In the paper we give some necessary facts from this theory. A lot of illustrative numerical examples are provided throughout the paper.

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