Convexity Hypothesis Research Articles

Approximation of $$L^\infty $$ functionals with generalized Orlicz norms

AbstractThe aim of this paper is to deal with the asymptotics of generalized Orlicz norms when the lower growth rate tends to infinity. We generalize results proven by Bertazzoni, Harjulehto and Hästö in Journ. of Math. Anal. and Appl. (2024) for integral type energies (in generalized Orlicz spaces), considering milder convexity assumptions. $$\Gamma $$ Γ -convergence results and related representation theorems in terms of $$L^\infty $$ L ∞ functionals are proven. The convexity hypotheses are completely removed in the variable exponent setting, thus extending the results in Eleuteri-Prinari in Nonlinear Anal. Real. World Appl. (2021) and Prinari-Zappale in JOTA (2020).

Double hedonic price-characteristics frontier estimation for IoT service providers in the industry 5.0 era: A nonconvex perspective accommodating ratios

The advent of advanced digital technologies, including the Internet of Things (IoT), image processing, artificial intelligence (AI), blockchain, robotics and cognitive computing that have been embedded in Industry 5.0, is considerably improving the sustainability, resilience, and human-centric performance of industrial organizations. Despite the increasing use of Industry 5.0 technologies in smart product platforming in industrial organizations, a critical issue remains how to assess the providers/suppliers of such technologies in highly competitive markets to fulfil personalized products and services. Following Lancaster's characteristics approach to consumer theory, in this study we contribute to assess digital technologies service providers in the Industry 5.0 era by focusing on both theoretical and empirical evidence inquiring about the convexity of conventional nonparametric frontier estimation methods. To do so, a nonparametric double frontier estimation of the hedonic price characteristics relation is developed from both the buyer's and seller's perspectives. Moreover, a separable directional distance function-based optimization model is developed for the efficiency estimation. Furthermore, a comparable estimation of the convex and nonconvex hedonic price function is proposed. We also explicitly test the impact of convexity in evaluating the efficiency of IoT service providers in the Industry 5.0 context. In this study, we also show that the hypothesis of convexity in assessing the efficiency of IoT service providers is rejected using the Li-test comparing entire densities in the case of the seller's perspective without ratio data. Differences are less pronounced for the buyer's perspective and in the case with ratio data.

Monotonicity and the Precision of Program Analysis

It is widely known that the precision of a program analyzer is closely related to intensional program properties, namely, properties concerning how the program is written. This explains, for instance, the interest in code obfuscation techniques, namely, tools explicitly designed to degrade the results of program analysis by operating syntactic program transformations. Less is known about a possible relation between what the program extensionally computes, namely, its input-output relation, and the precision of a program analyzer. In this paper we explore this potential connection in an effort to isolate program fragments that can be precisely analyzed by abstract interpretation, namely, programs for which there exists a complete abstract interpretation. In the field of static inference of numeric invariants, this happens for programs, or parts of programs, that manifest a monotone (either non-decreasing or non-increasing) behavior. We first formalize the notion of program monotonicity with respect to a given input and a set of numerical variables of interest. A sound proof system is then introduced with judgments specifying whether a program is monotone relatively to a set of variables and a set of inputs. The interest in monotonicity is justified because we prove that the family of monotone programs admits a complete abstract interpretation over a specific class of non-trivial numerical abstractions and inputs. This class includes all non-relational abstract domains that refine interval analysis (i.e., at least as precise as the intervals abstraction) and that satisfy a topological convexity hypothesis.

Periodic solutions for second-order even and noneven Hamiltonian systems

AbstractIn this paper, we consider the second-order Hamiltonian system $$ \ddot{x}+V^{\prime}(x)=0,\quad x\in \mathbb{R}^{N}. $$ x ¨ + V ′ ( x ) = 0 , x ∈ R N . We use the monotonicity assumption introduced by Bartsch and Mederski (Arch. Ration. Mech. Anal. 215:283–306, 2015). When V is even, we can release the strict convexity hypothesis, which is used by Bartsch and Mederski combined with the monotonicity assumption. When V is noneven, we weaken the strict convexity assumption and introduce another hypothesis (see (V10)). Then in both cases, we can build the homomorphism between the Nehari manifold and the unit sphere of some suitable space. Using the Nehari manifold method introduced by Szulkin (J. Funct. Anal. 257:3802–3822 2009), we prove the existence of T-periodic solutions with minimal period T.

A necessary condition for extremality of solutions to autonomous obstacle problems with general growth

Let us consider the autonomous obstacle problem minv∫ΩF(Dv(x))dxon a specific class of admissible functions, where we suppose the Lagrangian satisfies proper hypotheses of convexity and superlinearity at infinity. Our aim is to find a necessary condition for the extremality of the solution, which exists and it is unique, thanks to a primal–dual formulation of the problem. The proof is based on classical arguments of Convex Analysis and on Calculus of Variations’ techniques.

Optimality conditions and duality for nonsmooth multiobjective semi-infinite programming problems with vanishing constraints on Hadamard manifolds

This article is concerned with nonsmooth multiobjective semi-infinite programming problems with vanishing constraints in the setting of Hadamard manifolds (abbreviated as, (NMSIPVC)). We present the Abadie constraint qualification (abbreviated as, (ACQ)) and a modified version of (ACQ), namely, (NMSIPVC)-tailored Abadie constraint qualification (abbreviated as, (NMSIPVC-ACQ)), for (NMSIPVC). The Karush-Kuhn-Tucker (abbreviated as, KKT) type necessary criteria of optimality for (NMSIPVC) are derived by employing (ACQ) and (NMSIPVC-ACQ). Moreover, sufficient criteria of optimality for (NMSIPVC) are derived under generalized geodesic convexity hypotheses and certain mild restrictions on the index sets. Further, we formulate Wolfe type and Mond-Weir type dual models related to (NMSIPVC) and derive several duality results that relate (NMSIPVC) and the corresponding dual models. Non-trivial numerical examples are incorporated to demonstrate the validity of the results established in this paper. To the best of our knowledge, this is for the first time that constraint qualifications, optimality conditions, and duality results have been investigated for (NMSIPVC) in the setting of Hadamard manifolds.

Pareto Efficiency Criteria and Duality for Multiobjective Fractional Programming Problems with Equilibrium Constraints on Hadamard Manifolds

This article deals with multiobjective fractional programming problems with equilibrium constraints in the setting of Hadamard manifolds (abbreviated as MFPPEC). The generalized Guignard constraint qualification (abbreviated as GGCQ) for MFPPEC is presented. Furthermore, the Karush–Kuhn–Tucker (abbreviated as KKT) type necessary criteria of Pareto efficiency for MFPPEC are derived using GGCQ. Sufficient criteria of Pareto efficiency for MFPPEC are deduced under some geodesic convexity hypotheses. Subsequently, Mond–Weir and Wolfe type dual models related to MFPPEC are formulated. The weak, strong, and strict converse duality results are derived relating MFPPEC and the respective dual models. Suitable nontrivial examples have been furnished to demonstrate the significance of the results established in this article. The results derived in the article extend and generalize several notable results previously existing in the literature. To the best of our knowledge, optimality conditions and duality for MFPPEC have not yet been studied in the framework of manifolds.

On directionally differentiable multiobjective programming problems with vanishing constraints

In this paper, a class of directionally differentiable multiobjective programming problems with inequality, equality and vanishing constraints is considered. Under both the Abadie constraint qualification and the modified Abadie constraint qualification, the Karush–Kuhn–Tucker type necessary optimality conditions are established for such nondifferentiable vector optimization problems by using the nonlinear version Gordan theorem of the alternative for convex functions. Further, the sufficient optimality conditions for such directionally differentiable multiobjective programming problems with vanishing constraints are proved under convexity hypotheses. Furthermore, vector Wolfe dual problem is defined for the considered directionally differentiable multiobjective programming problem vanishing constraints and several duality theorems are established also under appropriate convexity hypotheses.

On proper minimality in set optimization

AbstractThe aim of this paper is to extend some notions of proper minimality from vector optimization to set optimization. In particular, we focus our attention on the concepts of Henig and Geoffrion proper minimality, which are well-known in vector optimization. We introduce a generalization of both of them in set optimization with finite dimensional spaces, by considering also a special class of polyhedral ordering cone. In this framework, we prove that these two notions are equivalent, as it happens in the vector optimization context, where this property is well-known. Then, we study a characterization of these proper minimal points through nonlinear scalarization, without considering convexity hypotheses.

Frequency criteria for exponential stability

Abstract We discuss some frequency-domain criteria for the exponential stability of nonlinear feedback systems based on Popov’s hyperstability theorem. The main results are new bounds on convergence rates for perturbations of the damped harmonic oscillator, including the Hessian-damped case, when the perturbations satisfy certain sectoriality or convexity hypotheses. We also include a simplified proof of Popov’s theorem in its general form, which has the virtue that it often applies at the margins of stability when other criteria typically degenerate.

New Notions of Proper Efficiency in Set Optimization with the Set Criterion

In this paper, we introduce new notions of proper efficiency in the sense of Henig for a set optimization problem by using the set criterion of solution. The relationships between them are studied. Also, we compare these concepts with the homologous ones given by considering the vector criterion. Finally, a Lagrange multiplier rule for Henig proper solutions of a set optimization problem with a cone constraint is obtained under convexity hypotheses. Illustrative examples are also given.

Minty Variational Principle for Nonsmooth Interval-Valued Vector Optimization Problems on Hadamard Manifolds

This article deals with the classes of approximate Minty- and Stampacchia-type vector variational inequalities on Hadamard manifolds and a class of nonsmooth interval-valued vector optimization problems. By using the Clarke subdifferentials, we define a new class of functions on Hadamard manifolds, namely, the geodesic LU-approximately convex functions. Under geodesic LU-approximate convexity hypothesis, we derive the relationship between the solutions of these approximate vector variational inequalities and nonsmooth interval-valued vector optimization problems. This paper extends and generalizes some existing results in the literature.

Modified limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm for unconstrained optimization problem

The use of the self-scaling Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is very efficient for the resolution of large-scale optimization problems, in this paper, we present a new algorithm and modified the self-scaling BFGS algorithm. Also, based on noticeable non-monotone line search properties, we discovered and employed a new non-monotone idea. Thereafter first, an updated formula is exhorted to the convergent Hessian matrix and we have achieved the secant condition, second, we established the global convergence properties of the algorithm under some mild conditions and the objective function is not convexity hypothesis. A promising behavior is achieved and the numerical results are also reported of the new algorithm.

Optimality Conditions for Multiobjective Programming under Generalized Uniform (V, η)-type I Symmetrical Invexity

Multi-objective optimization is a major branch of the optimization category. On the basis of (V, η)-I symmetric invariant convex functions, the concept of generalized uniform (V, η)-I symmetric invariant convex function is proposed. Which is generalized uniform (V, η)-I type. Under the new generalized convexity hypothesis, several optimality sufficient conditions for multi-objective programming are proved.

Compactification in optimal control of McKean‐Vlasov stochastic differential equations

AbstractWe study existence and approximation of optimal controls for systems governed by McKean‐Vlasov stochastic differential equations. It is well known in simple examples that in the absence of convexity conditions, the strict control problem has no optimal solution. The compactification of the set of such strict admissible controls leads to measure valued controls called relaxed controls. The space of relaxed controls enjoys nice topological properties. We prove that under pathwise uniqueness of solutions of the state equation, the relaxed state process is continuous with respect to the control variable. This means that the relaxed and strict control problems have the same value function. Moreover, we show, under merely continuity of the coefficients, that an optimal control exists in the space of relaxed controls. Under additional convexity hypothesis, we show that the optimal relaxed control is a strict control. These two results extend known results to general nonlinear MVSDEs, under minimal assumptions on the coefficients.

РОЗВ’ЯЗАННЯ ЗАДАЧІ ОПТИМАЛЬНОГО КЕРУВАННЯ ПОТУЖНІСТЮ ЕЛЕКТРОЕНЕРГЕТИЧНОГО КОМПЛЕКСУ СУДНА

Для забезпечення ефективного керування судновою електроенергетичною системою (СЕЕС) при стабілізації руху судна в умовах, які потребують використання динамічного компенсування збурень, зокрема хвиль та вітру, зазвичай реалізують метод динамічної компенсації. При цьому умови вітрових, хвильових навантажень і течій не обмежені. З огляду на тоннаж і конструктивні особливості суден, створення системи керування, інваріантної до збурень, є вкрай складним завданням. Взагалі, глобальна задача оптимізації включає задачі оптимального регулювання пропульсивного комплексу, оптимального керування дизель-генераторами, оптимальне керування СЕЕС за критерієм мінімальної узагальненої роботи та квазіоптимальне регулювання в СЕЕС. Особливості експлуатації пропульсивного комплексу судна значним чином змінюють навантаження на гребні електродвигуни (судновий пропульсивний комплекс), що призводить до необхідності створення спеціальних систем маневрування та керування СЕЕС. Перевагою електричних пропульсивних комплексів, на відміну від дизельних, є можливість гнучкого перерозподілу потужностей між приводними двигунами, висока точність та швидкодія, що є найважливішим фактором для суден з динамічним позиціюванням. При цьому основною задачею такої системи є оптимальне керування за критерієм мінімальної узагальненої роботи. У роботі розглянуто задачу оптимізації роботи дизель-генераторів при обмеженні з боку сталості упору гвинта. Показано, що для забезпечення сталості упору гвинта треба використовувати методи оптимального керування за критерієм максимуму швидкодії. Також проаналізовано гіпотезу опуклості функціонала цілі і керованості обмежень, яка дозволяє реалізувати принцип максимуму Понтрягіна та принцип Белмана і визначити залежність якості регулювання від ресурсу керування. Розв’язання задачі оптимального керування СЕЕС відповідає критерію мінімальної узагальненої роботи, а використання принципу Белмана дозволяє реалізувати в подальшому адаптивний регулятор, що забезпечує мінімальні витрати за керуванням.

Optimality Conditions for a Nonsmooth Uncertain Multiobjective Programming Problem

This note is devoted to the investigation of optimality conditions for robust approximate quasi weak efficient solutions to a nonsmooth uncertain multiobjective programming problem (NUMP). Firstly, under the extended nonsmooth Mangasarian–Fromovitz constrained qualification assumption, the optimality necessary conditions of robust approximate quasi weak efficient solutions are given by using an alternative theorem. Secondly, a class of generalized convex functions is introduced to the problem (NUMP), which is called the pseudoquasi-type-I function, and its existence is illustrated by a concrete example. Finally, under the pseudopseudo-type-I generalized convexity hypothesis, the optimality sufficient conditions for robust approximate quasi weak efficient solutions to the problem (NUMP) are established.

An exact l1 penalty function method for a multitime control optimization problem with data uncertainty

SummaryIn this article, we study the robust solution set of a multitime first‐order PDE constrained control optimization problem with data uncertainty (MCOPU) and an unconstrained multitime control optimization problem (MCOPU)ρ. We establish the equivalence between a robust optimal solution of (MCOPU) and a robust minimizer of (MCOPU)ρ under the appropriate assumptions. Furthermore, we define the uncertain Lagrange functional for (MCOPU) and show the relationship between the robust solution set of (MCOPU) and (MCOPU)ρ under the convexity hypothesis of uncertain Lagrange functional. Moreover, we present some nontrivial examples to illustrate the established results.

Optimality conditions for approximate proper solutions in multiobjective optimization with polyhedral cones

In this paper, we provide optimality conditions for approximate proper solutions of a multiobjective optimization problem, whose feasible set is given by a cone constraint and the ordering cone is polyhedral. A first class of optimality conditions is given by means of a nonlinear scalar Lagrangian and the second kind through a linear scalarization technique, under generalized convexity hypotheses, that lets us derive a Kuhn–Tucker multiplier rule.

Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function

In this paper, a nonconvex vector optimization problem with multiple interval-valued objective function and both inequality and equality constraints is considered. The functions constituting it are not necessarily differentiable, but they are $ E $-differentiable. The so-called $ E $-Karush-Kuhn-Tucker necessary optimality conditions are established for the considered $ E $-differentiable vector optimization problem with the multiple interval-valued objective function. Also the sufficient optimality conditions are derived for such interval-valued vector optimization problems under appropriate (generalized) $ E $-convexity hypotheses.