arXiv subjects

Vsevolod I. Ivanov

Publications and source records attributed to Vsevolod I. Ivanov.

9 recordsLinked to original sources

arXiv2026

Optimality conditions with cone-generalized convex vector functions

The object of investigation in this paper are generalized convex vector nonlinear programming problems with cone constraints. We prove that a quasiconvex vector function is pseudoconvex with respect to the same cone if and only if every vector critical point of the function is a weak global minimizer, a result, which is a natural extension of a known characterization of pseudoconvex scalar functions. We introduce a new notion of a Fréchet differentiable pseudoconvex problem. We derive that a problem with quasiconvex vector data is pseudoconvex if and only if every Fritz John vector critical point is a weakly efficient global solution. Thus, we generalize a lot of previous optimality conditions, concerning the scalar case and the multiobjective Paretian one. We also introduce the notion of a Fritz John cone-pseudoinvex vector problem. We prove that a problem with cone constraints is Fritz John pseudoinvex if and only if every vector critical point of Fritz John type is a weak global minimizer with respect to the considered cone..

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arXiv2025

Characterizations of Pseudolinear and Semi-strictly Quasilinear Functions

In this paper we obtain several new complete characterizations of pseudolinear functions. Two of the results are of first-order and one is derivative free. All results are derived in terms of the Clarke-Rockafellar subdifferential. Additionally, we prove a characterization of the semi-strictly quasilinear functions. It is similar to the derivative free characterization of the pseudolinear functions. We also find the conditions such that a semi-strictly quasilinear function become pseudolinear.

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arXiv2025

Multiobjective Programming and Weighting Scalar Problem

In this work is discussed the paper by Osuna-Gomez, Beato-Moreno and Rufian-Lizana, Generalized convexity in multiobjective programming: J. Math. Anal. Appl., v. 233 (1999) pp. 205--220. We point out an error in the proofs of main Theorems 2.4 and 3.6 in this reference. Then we derive correct proofs, which are based on a different approach, because the scheme of the proofs of the authors is not applicable. As a result of the discussion two additional theorems are obtained such that the scheme of the proofs of Theorems 2.4 and 3.6 in the mentioned reference works for them.

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arXiv2024

First and Second Order Necessary and Sufficient Optimality Conditions of Fritz John Type for Vector Problems over Cones

In this paper, we obtain a new proof of Fritz John necessary optimality conditions for vector problems applying Kakutani fixed point theorem and Hadamard directional derivative. We also derive a similar proof of second-order Fritz John necessary optimality conditions. Sufficient conditions for weak global efficiency with generalized convex functions and local efficiency are provided.

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arXiv2018

Second-order optimality conditions and Lagrange multiplier characterizations of the solution set in quasiconvex programming

Second-order optimality conditions for vector nonlinear programming problems with inequality constraints are studied in this paper. We introduce a new second-order constraint qualification, which includes Mangasarian-Fromovitz constraint qualification as a particular case. We obtain necessary and sufficient conditions for weak efficiency of problems with a second-order pseudoconvex vector objective function and quasiconvex constraints. We also derive Lagrange multiplier characterizations of the solution set of a scalar problem with a second-order pseudoconvex objective function and quasiconvex inequality constraints, provided that one of the solutions and the Lagrange multipliers in the Karush-Kuhn-Tucker conditions are known. At last, we introduce a notion of a second-order KKT-pseudoconvex problem with inequality constraints. We derive sufficient and also necessary conditions for efficiency of second-order KKT-pseudoconvex problems. Three examples are presented.

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arXiv2016

Higher-order optimality conditions with an arbitrary non-differentiable function

In this paper, we introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper extended real function. This derivative is harmonized with the classical higher-order Fréchet directional derivative in the sense that both derivatives of the same order coincide if the last one exists. We obtain necessary and sufficient conditions of order $n$ ($n$ is a positive integer) for a local minimum and isolated local minimum of order $n$ of the given function in terms of these derivatives and subdifferentials. We do not require any restrictions on the function in our results. A notion of a higher-order critical direction is introduced. It is applied in the characterizations of the isolated local minimum of order $n$. Higher-order invex functions are defined. They are the largest class such that our necessary conditions for local minima are sufficient for global one. We compare our results with some previous ones. As an application, we improve a result due to V. F. Demyanov, showing that the condition introduced by this author is a complete characterization of isolated local minimizers of order $n$.

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arXiv2014

Second-Order Optimality Conditions in Cone-Constrained Vector Optimization with Arbitrary Nondifferentiable Functions

In this paper, we introduce a new second-order directional derivative and a second-order subdifferential of Hadamard type for an arbitrary nondifferentiable function. We derive several second-order optimality conditions for a local and a global minimum and an isolated local minimum of second-order in unconstrained optimization. In particular, we obtain two types results with generalized convex functions. We also compare our conditions with the results of the recently published paper [Bednavrik, D., Pastor, K.: On second-order conditions in unconstrained optimization. Math. Program. Ser A, {\bf 113}, 283--291 (2008)] and a lot of other works, published in high level journals, and prove that they are particular cases of our necessary and sufficient ones. We prove that the necessary optimality conditions concern more functions than the lower Dini directional derivative, even the optimality conditions with the last derivative can be applied to a function, which does not belong to some special class. At last, we apply our optimality criteria for unconstrained problems to derive necessary and sufficient optimality conditions for the cone-constrained vector problems.

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arXiv2014

Second-Order Karush-Kuhn-Tucker Optimality Conditions for Vector Problems with Continuously Differentiable Data and Second-Order Constraint Qualifications

Some necessary and sufficient optimality conditions for inequality constrained problems with continuously differentiable data were obtained in the papers [I. Ginchev and V.I. Ivanov, Second-order optimality conditions for problems with C$\sp{1}$ data, J. Math. Anal. Appl., v. 340, 2008, pp. 646--657], [V.I. Ivanov, Optimality conditions for an isolated minimum of order two in C$\sp{1}$ constrained optimization, J. Math. Anal. Appl., v. 356, 2009, pp. 30--41] and [V. I. Ivanov, Second- and first-order optimality conditions in vector optimization, Internat. J. Inform. Technol. Decis. Making, 2014, DOI: 10.1142/S0219622014500540]. In the present paper, we continue these investigations. We obtain some necessary optimality conditions of Karush--Kuhn--Tucker type for scalar and vector problems. A new second-order constraint qualification of Zangwill type is introduced. It is applied in the optimality conditions.

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arXiv2013

Optimality Conditions in Vector Optimization

In this paper we obtain second- and first-order optimality conditions of Kuhn-Tucker type and Fritz John one for weak efficiency in the vector problem with inequality constraints. In the necessary conditions we suppose that the objective function and the active constraints are continuously differentiable. We introduce notions of KTSP-invex problem and second-order KTSP-invex one. We obtain that the vector problem is (second-order) KTSP-invex if and only if for every triple $(\bar x,\barλ,\barμ)$ with Lagrange multipliers $\barλ$ and $\barμ$ for the objective function and constraints respectively, which satisfies the (second-order) necessary optimality conditions, the pair $(\bar x,\barμ)$ is a saddle point of the scalar Lagrange function with a fixed multiplier $\barλ$. We introduce notions second-order KT-pseudoinvex-I, second-order KT-pseudo\-invex-II, second-order KT-invex problems. We prove that every second-order Kuhn-Tucker stationary point is a weak global Pareto minimizer (global Pareto minimizer) if and only if the problem is second-order KT-pseudoinvex-I (KT-pseudoinvex-II). It is derived that every second-order Kuhn Tucker stationary point is a global solution of the weighting problem if and only if the vector problem is second-order KT-invex.

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