Search

The physical regions (domains or basins) within the molecular structure are open systems that exchange charge between them and, consequently, house a fractional number of electrons (net charge). The natural framework describing the quantum states for these domains is the density matrix (DM) in its grand-canonical version, which corresponds to a convex expansion into a set of basis states of an integer number of electrons. In this report, it is shown that the solution for these quantities is supported by the DM expansion into three states of different numbers of particles: the neutral and two (edge) ionic states. The states and the average number of particles in the domains (fractional occupation population) are determined by the coefficients of the expansion in terms of the fundamental transference magnitudes, revealing the donor/acceptor character of the domains by which the quantum accessible states are discussed. [ABSTRACT FROM AUTHOR]

Part 2 of this article is devoted to the analysis of a new class of combination rules on belief functions generated by conditioning w.r.t. random sets and their linear convex combinations. Each such rule obeys two requirements: the consensus requirement with the conjunction rule means that this rule coincides with Dempster's rule on non-conflicting sources of information, and the consensus requirement with a mixture rule means that the result must be a specialization of their mixture combination. We show that this new class of combination rules is closed under convex combinations of such rules and their compositions. This allows us to implement the conflict management using the introduced functionals for measuring the specialization relation on belief functions. We also introduce similar rules for idempotent combination rules on credal sets obeying the similar consensus requirements and study the results obtained using various examples. [ABSTRACT FROM AUTHOR]

We explore and construct an enlarged subdifferential for weakly convex functions. The resulting object turns out to be continuous with respect to both the function argument and the enlargement parameter. We carefully analyze connections with other constructs in the literature and particularize to the weakly convex setting well-known variational principles. By resorting to the new enlarged subdifferential, we provide an algorithmic pattern of descent for weakly convex minimization. Under minimal assumptions, we show subsequential convergence to a critical point. Links with difference-of-convex functions algorithms and criticality conditions are also discussed. [ABSTRACT FROM AUTHOR]

Conic linear programs in locally convex Hausdorff topological vector spaces are addressed in this paper. Solution existence for the dual problem, as well as solution existence for the primal problem, and strong duality, are proved under minimal regularity assumptions. Namely, to get the results and a Farkas-type theorem for infinite-dimensional conic linear inequalities, we employ the generalized Slater condition either for the primal problem or for the dual problem, as well as proper separation and the concept of quasi-regularity of convex sets. Illustrative examples are presented. [ABSTRACT FROM AUTHOR]

We study the set of intrinsic metrics on a given graph. This is a convex compact set and it carries a natural order. We investigate existence of largest elements with respect to this order. We show that the only locally finite graphs which admit a largest intrinsic metric are certain finite star graphs. In particular, all infinite locally finite graphs do not admit a largest intrinsic metric. For infinite graphs which are not locally finite the set of intrinsic metrics may be trivial as we show by an example. Moreover, we give a characterization for the existence of intrinsic metrics with finite balls for weakly spherically symmetric graphs. [ABSTRACT FROM AUTHOR]

In this paper, we study the hyperbolic and parabolic strip deformations of ideal (possibly once‐punctured) hyperbolic polygons whose vertices are decorated with horoballs. We prove that the interiors of their arc complexes parametrise the open convex set of all uniformly lengthening infinitesimal deformations of the decorated hyperbolic metrics on these surfaces, motivated by the work of Danciger–Guéritaud–Kassel. We also give a version of this result for the undecorated ideal polygons and once‐punctured ideal polygons. [ABSTRACT FROM AUTHOR]

Adverse selection is a version of the principal-agent problem that includes monopolist nonlinear pricing, where a monopolist with known costs seeks a profit-maximizing price menu facing a population of potential consumers whose preferences are known only in the aggregate. For multidimensional spaces of agents and products, Rochet and Choné (Econometrica, 1998) reformulated this problem as a concave maximization over the set of convex functions, by assuming agent preferences combine bilinearity in the product and agent parameters with a quasilinear sensitivity to prices. We characterize solutions to this problem by identifying a dual minimization problem. This duality allows us to reduce the solution of the square example of Rochet–Choné to a novel free boundary problem, giving the first analytical description of an overlooked market segment. [ABSTRACT FROM AUTHOR]

This paper considers the event‐triggered adaptive fuzzy bipartite containment control problem for switched nonlinear multi‐agent systems (MASs) with actuator attacks. To handle the completely unknown nonlinear functions in the considered system, fuzzy logic systems (FLSs) are applied to approximate the unknown nonlinear functions. In addition, an adaptive bipartite containment control approach is designed to deal with the actuator attacks, which can reduce the burden of communication. According to the designed adaptive bipartite containment control scheme, it is shown that all the signals are semi‐globally uniformly ultimately bounded (SGUUB), and all the outputs of the followers achieve the bipartite consensus and finally enter the convex set spanned by leaders' in the presence of actuator attacks. Eventually, a simulation example is presented to illustrate the availability of the developed strategy. [ABSTRACT FROM AUTHOR]

Polyhedral convex set optimization problems are the simplest optimization problems with set-valued objective function. Their role in set optimization is comparable to the role of linear programs in scalar optimization. Vector linear programs and multiple objective linear programs provide proper subclasses. In this article, we choose a solution concept for arbitrary polyhedral convex set optimization problems out of several alternatives, show existence of solutions and characterize the existence of solutions in different ways. Two known results are obtained as particular cases, both with proofs being easier than the original ones: The existence of solutions of bounded polyhedral convex set optimization problems and a characterization of the existence of solutions of vector linear programs. [ABSTRACT FROM AUTHOR]

In this paper, we consider the closed convex sets with full Pareto-like relative boundary and call them Pareto bordered sets. We gradually characterize such sets by paying some special attention, at first, to those Pareto bordered sets which contain no lines and then to the arbitrary ones. We also provide some additional sufficient requirements on the representatives of a family of Pareto bordered sets such that their intersection remains Pareto bordered. [ABSTRACT FROM AUTHOR]

The Frank–Wolfe (FW) method, which implements efficient linear oracles that minimize linear approximations of the objective function over a fixed compact convex set, has recently received much attention in the optimization and machine learning literature. In this paper, we propose a new FW-type method for minimizing a smooth function over a compact set defined as the level set of a single difference-of-convex function, based on new generalized linear-optimization oracles (LO). We show that these LOs can be computed efficiently with closed-form solutions in some important optimization models that arise in compressed sensing and machine learning. In addition, under a mild strict feasibility condition, we establish the subsequential convergence of our nonconvex FW-type method. Since the feasible region of our generalized LO typically changes from iteration to iteration, our convergence analysis is completely different from those existing works in the literature on FW-type methods that deal with fixed feasible regions among subproblems. Finally, motivated by the away steps for accelerating FW-type methods for convex problems, we further design an away-step oracle to supplement our nonconvex FW-type method, and establish subsequential convergence of this variant. Numerical results on the matrix completion problem with standard datasets are presented to demonstrate the efficiency of the proposed FW-type method and its away-step variant. [ABSTRACT FROM AUTHOR]

Consider the nonlinear pseudo-monotone equations over a nonempty closed convex set. A spectral conjugate gradient projection method with the inertial factor is proposed for solving the problem under discussion. Following the projection strategy, we prove that the sequence of spectral parameters is bounded. The search direction generated by the algorithm satisfies the sufficient descent condition and possesses trust region property at each iteration. Under some mild conditions, the global convergence of the proposed method is established without the Lipschitz continuity assumption. Under some standard assumptions, we also establish the linear convergence rate of our method. Preliminary numerical results on constrained nonlinear monotone and pseudo-monotone equations demonstrate the efficiency of the proposed method. Furthermore, to highlight its applicability, we extend our method to deal with logistic regression problems. [ABSTRACT FROM AUTHOR]

In this paper, we explore questions regarding the Minkowski sum of the boundaries of convex sets. Motivated by a question suggested to us by V. Milman regarding the volume of ∂K +∂T where K and T are convex bodies, we prove sharp volumetric lower bounds for the Minkowski average of the boundaries of sets with connected boundary, as well as some related results. [ABSTRACT FROM AUTHOR]

In this paper, we study online diminishing return submodular (DR-submodular for short) maximization in the bandit setting. Our focus is on problems where the reward functions can be non-monotone, and the constraint set is a general convex set. We first present the Single-sampling Non-monotone Frank-Wolfe algorithm. This algorithm only requires a single call to each reward function, and it computes the stochastic gradient to make it suitable for large-scale settings where full gradient information might not be available. We provide an analysis of the approximation ratio and regret bound of the proposed algorithm. We then propose the Bandit Online Non-monotone Frank-Wolfe algorithm to adjust for problems in the bandit setting, where each reward function returns the function value at a single point. We take advantage of smoothing approximations to reward functions to tackle the challenges posed by the bandit setting. Under mild assumptions, our proposed algorithm can reach 1 4 (1 - min x ∈ P ′ ‖ x ‖ ∞) -approximation with regret bounded by O (T 5 min { 1 , γ } + 5 6 min { 1 , γ } + 5 ) , where the positive parameter γ is related to the "safety domain" P ′ . To the best of our knowledge, this is the first work to address online non-monotone DR-submodular maximization over a general convex set in the bandit setting. [ABSTRACT FROM AUTHOR]

Let Y be a real vector metric space and K ⊂ Y be a closed convex cone such that K ∩ (− K) = {0}. We prove that a convex compact-valued map F : ℝ → 2Y ∖ {∅} is K-continuous and K-additive if and only if there are non-empty convex compact sets A, B ⊂ Y such that 0 ∈ A − B ⊂ K and F is equal "modulo K" to the continuous set-valued map G (t) = t A , t ≥ 0 , t B , t < 0. Next, we use this result to characterize convex compact-valued maps F : ℝN → 2Y ∖ {∅}. [ABSTRACT FROM AUTHOR]

This paper is focussed on some properties of paramonotone operators on Banach spaces and their application to certain feasibility problems for convex sets in a Hilbert space and convex systems in the Euclidean space. In particular, it shows that operators that are simultaneously paramonotone and bimonotone are constant on their domains, and this fact is applied to tackle two particular situations. The first one, closely related to simultaneous projections, deals with a finite amount of convex sets with an empty intersection and tackles the problem of finding the smallest perturbations (in the sense of translations) of these sets to reach a nonempty intersection. The second is focussed on the distance to feasibility; specifically, given an inconsistent convex inequality system, our goal is to compute/estimate the smallest right-hand side perturbations that reach feasibility. We advance that this work derives lower and upper estimates of such a distance, which become the exact value when confined to linear systems. [ABSTRACT FROM AUTHOR]

Objectives: Evaluate microcalcification detectability in digital breast tomosynthesis (DBT) and synthetic 2D mammography (SM) for different acquisition setups using a virtual imaging trial (VIT) approach. Materials and methods: Medio-lateral oblique (MLO) DBT acquisitions on eight patients were performed at twice the automatic exposure controlled (AEC) dose. The noise was added to the projections to simulate a given dose trajectory. Virtual microcalcification models were added to a given projection set using an in-house VIT framework. Three setups were evaluated: (1) standard acquisition with 25 projections at AEC dose, (2) 25 projections with a convex dose distribution, and (3) sparse setup with 13 projections, every second one over the angular range. The total scan dose and angular range remained constant. DBT volume reconstruction and synthetic mammography image generation were performed using a Siemens prototype algorithm. Lesion detectability was assessed through a Jackknife-alternative free-response receiver operating characteristic (JAFROC) study with six observers. Results: For DBT, the area under the curve (AUC) was 0.97 ± 0.01 for the standard, 0.95 ± 0.02 for the convex, and 0.89 ± 0.03 for the sparse setup. There was no significant difference between standard and convex dose distributions (p = 0.309). Sparse projections significantly reduced detectability (p = 0.001). Synthetic images had a higher AUC with the convex setup, though not significantly (p = 0.435). DBT required four times more reading time than synthetic mammography. Discussion: A convex setup did not significantly improve detectability in DBT compared to the standard setup. Synthetic images exhibited a non-significant increase in detectability with the convex setup. Sparse setup significantly reduced detectability in both DBT and synthetic mammography. Clinical relevance statement: This virtual imaging trial study allowed the design and efficient testing of different dose distribution trajectories with real mammography images, using a dose-neutral protocol. Key Points: • In DBT, a convex dose distribution did not increase the detectability of microcalcifications compared to the current standard setup but increased detectability for the SM images. • A sparse setup decreased microcalcification detectability in both DBT and SM images compared to the convex and current clinical setups. • Optimal microcalcification cluster detection in the system studied was achieved using either the standard or convex dose setting, with the default number of projections. [ABSTRACT FROM AUTHOR]

In this paper, we establish several proper separation theorems for an element and a convex set and for two convex sets in terms of their quasi-relative interiors. Then, we prove that the separation theorem given by [Cammaroto, F and B Di Bella (2007). On a separation theorem involving the quasi-relative. Proceedings of the Edinburgh Mathematical Society, 50(3), 605–610] in Theorem 2.5, is in fact a proper separation theorem for two convex sets in which the classical interior is replaced by the quasi-relative interior. Besides, we extend some known results in the literature, such as [Adán, M and V Novo (2004). Proper efficiency in vector optimization on real linear spaces. Journal of Optimization Theory and Applications, 121, 515–540] in Theorem 2.1 and [Edwards, R (1965). Functional Analysis: Theory and Applications. New York: Reinhart and Winston] in Corollary 2.2.2, through the quasi-relative interior and the quasi-interior, respectively. As an application, we provide Karush–Kuhn–Tucker multipliers for quasi-relative solutions of vector optimization problems. Several examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

We continue the study of the properties of the V -Moreau envelope and generalized (f , λ) -projection that we started in [5]. In this paper, we study the differentiability and the regularity of the V -Moreau envelope and the Hölder continuity of the generalized (f , λ) -projection. Our results extend many existing results from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. Even on Hilbert spaces and for convex functions and sets, we derived new results. [ABSTRACT FROM AUTHOR]

In this paper, we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank 2 if and only if each open face in the ideal boundary has dimension at most one. We also introduce the “coarse Hilbert dimension” of a subset of a convex set and use it to characterize when a naive convex co-compact subgroup is word hyperbolic or relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank 2. [ABSTRACT FROM AUTHOR]

We investigate location problems where the optimal solution is found within the tropical convex hull of the given input points. Our initial focus is on geodesically star-convex sets, using the asymmetric tropical distance. We introduce the concept of tropically quasiconvex functions, which have sub-level sets with this shape, and are closely related to monotonic functions. Our findings demonstrate that location problems using tropically quasiconvex functions as distance measures will result in an optimal solution within the tropical convex hull of the input points. We also extend this result to cases where the input points are replaced with tropically convex sets. Finally, we explore the applications of our research in phylogenetics, highlighting the properties of consensus methods that arise from our class of location problems. [ABSTRACT FROM AUTHOR]

Training large machine learning models requires a distributed computing approach, with communication of the model updates being the bottleneck. For this reason, several methods based on the compression of updates were recently proposed using sparsification or quantization. However, none of the prior methods are able to learn the gradients, which renders them incapable of converging to the true optimum in the batch mode. In this work we propose a new method – DIANA – which resolves this issue via compression of gradient differences. We provide theory in the strongly convex and nonconvex settings that shows improved convergence rates, and use it to obtain the first convergence rate for the previously proposed method TernGrad. Finally, we provide theory to support non-smooth regularizers. [ABSTRACT FROM AUTHOR]

Incomplete cooperative games generalize the classical model of cooperative games by omitting the values of some of the coalitions. This allows for incorporating uncertainty into the model and studying the underlying games and possible payoff distributions based only on the partial information. In this paper, we conduct a systematic investigation of incomplete games, focusing on two important classes: positive and convex games. Regarding positivity, we generalize previous results from a special class of minimal incomplete games to a general setting. We characterize the non-extendability to a positive game by the existence of a certificate and provide a description of the set of positive extensions using its extreme games. These results also enable the construction of explicit formulas for several classes of incomplete games with special structures. The second part deals with convexity. We begin with the case of non-negative, minimal incomplete games. We establish the connection between incomplete games and the problem of completing partial functions and, consequently, provide a characterization of extendability and a full description of the set of symmetric convex extensions. This set serves as an approximation of the set of convex extensions. [ABSTRACT FROM AUTHOR]

We focus on the problem of finding the greatest element of the intersection of max-closed convex sets. For this purpose, we analyze the famous method of cyclic projections and show that, if this method is suitably initialized and applied to max-closed convex sets, it converges to the greatest element of their intersection. Moreover, we propose another projection method, called the decreasing projection, which turns out both theoretically and practically preferable to Euclidean projections in this particular case. Next, we argue that several known algorithms, such as Bellman-Ford and Floyd-Warshall algorithms for shortest paths or Gauss-Seidel variant of value iteration in Markov decision processes, can be interpreted as special cases of iteratively applying decreasing projections onto certain max-closed convex sets. Finally, we link decreasing projections (and thus also the aforementioned algorithms) to bounds consistency in constraint programming. [ABSTRACT FROM AUTHOR]

Some properties and applications of generalized convex sets and generalized convex functions in multidimensional real, complex, and hypercomplex spaces are described. [ABSTRACT FROM AUTHOR]

This study analyzes the conditional gradient method for constrained multiobjective optimization problems, also known as the Frank–Wolfe method. We assume that the objectives are continuously differentiable, and the constraint set is convex and compact. We employ an average-type nonmonotone line search, which takes the average of the recent objective function values. The asymptotic convergence properties without convexity assumptions on the objective functions are established. We prove that every limit point of the sequence of iterates that is obtained by the proposed method is a Pareto critical point. An iteration-complexity bound is provided regardless of the convexity assumption on the objective functions. The effectiveness of the suggested approach is demonstrated by applying it to several benchmark test problems. In addition, the efficiency of the proposed algorithm in generating approximations of the entire Pareto front is compared to the existing Hager–Zhang conjugate gradient method, the steepest descent method, the monotone conditional gradient method, and a nonmonotone conditional gradient method. In finding empirical comparison, we utilize two commonly used performance matrices—inverted generational distance and hypervolume indicators. [ABSTRACT FROM AUTHOR]

It is conjectured that if a finite set of points in the plane contains many collinear triples, then part of the set has a structure. We will show that under some combinatorial conditions, such pointsets have special configurations of triples, proving a case of Elekes' conjecture. Using the techniques applied in the proof, we show a density version of Jamison's theorem. If the number of distinct directions between many pairs of points of a point set in a convex position is small, then many points are on a conic. [ABSTRACT FROM AUTHOR]

We investigate a number of questions, problems, and conjectures related to geometric transversal theory. Among our results we disprove a conjecture of Bárány and Kalai regarding weak ε -nets for k-flats and convex sets in R d , and we prove a conjecture of Arocha, Bracho, and Montejano regarding a colorful version of the Goodman–Pollack–Wenger transversal theorem. We also investigate the connected components of the space of line transversals to pairwise disjoint convex sets in R 3 , and we extend a theorem of Karasev and Montejano regarding colorful intersections and k-transversals. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a novel T-S fuzzy parameter varying system with Markov jump, in which parameters depend not only on a Markov chain but also on linear parameter varying elements that take values in convex polytopic sets. Stable conditions and the gain-scheduling controller design method for this system are obtained. Applying Lyapunov function depending on the operation mode and full block S-procedure lemma, we obtain stochastic stabilization conditions. We find that this novel system has two distinct advantages. On the one hand, it inherits the advantages of traditional T-S fuzzy systems in handling nonlinear objects under the frame of T-S fuzzy systems; on the other, it obtains the advantages of dealing with time-varying characteristics from the point of linear parameter varying (LPV) systems. Finally, the theory results are illustrated via numerical simulation. [ABSTRACT FROM AUTHOR]

We present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Posing controls on the porous part of the medium (distributed control), we set up a convex minimization problem. The main objective of this article is to characterize an arbitrary control to be an optimal control. We establish a relation between the optimal control and the corresponding adjoint state. At first, we analyse the microscopic description of the controlled system, then we homogenised the system by rigorous two-scale convergence method and periodic unfolding method. [ABSTRACT FROM AUTHOR]

We consider robust empirical risk minimization (ERM), where model parameters are chosen to minimize the worst-case empirical loss when each data point varies over a given convex uncertainty set. In some simple cases, such problems can be expressed in an analytical form. In general the problem can be made tractable via dualization, which turns a min-max problem into a min-min problem. Dualization requires expertise and is tedious and error-prone. We demonstrate how CVXPY can be used to automate this dualization procedure in a user-friendly manner. Our framework allows practitioners to specify and solve robust ERM problems with a general class of convex losses, capturing many standard regression and classification problems. Users can easily specify any complex uncertainty set that is representable via disciplined convex programming (DCP) constraints. [ABSTRACT FROM AUTHOR]

There is a long-standing problem, posed by A.T.-M. Lau [ Fixed point theory and its applications , Academic Press, New York-London, 1976, pp. 121–129], whether left amenability is sufficient to ensure the existence of a common fixed point for every jointly weak^{\ast } continuous nonexpansive semigroup action on a nonempty weak^{\ast } compact convex set in a dual Banach space. In this note we discuss the current status of this problem and give a partial solution in the case of weak^{\ast } compact convex sets with the Radon–Nikodým property. [ABSTRACT FROM AUTHOR]

We consider the multilinear polytope defined as the convex hull of the set of binary points z, satisfying a collection of equations of the form z e = ∏ v ∈ e z v for all e ∈ E . The complexity of the facial structure of the multilinear polytope is closely related to the acyclicity degree of the underlying hypergraph. We obtain a polynomial-size extended formulation for the multilinear polytope of β -acyclic hypergraphs, hence characterizing the acyclic hypergraphs for which such a formulation can be constructed. [ABSTRACT FROM AUTHOR]

Stochastic (sub)gradient methods require step size schedule tuning to perform well in practice. Classical tuning strategies decay the step size polynomially and lead to optimal sublinear rates on (strongly) convex problems. An alternative schedule, popular in nonconvex optimization, is called geometric step decay and proceeds by halving the step size after every few epochs. In recent work, geometric step decay was shown to improve exponentially upon classical sublinear rates for the class of sharp convex functions. In this work, we ask whether geometric step decay similarly improves stochastic algorithms for the class of sharp weakly convex problems. Such losses feature in modern statistical recovery problems and lead to a new challenge not present in the convex setting: the region of convergence is local, so one must bound the probability of escape. Our main result shows that for a large class of stochastic, sharp, nonsmooth, and nonconvex problems a geometric step decay schedule endows well-known algorithms with a local linear (or nearly linear) rate of convergence to global minimizers. This guarantee applies to the stochastic projected subgradient, proximal point, and prox-linear algorithms. As an application of our main result, we analyze two statistical recovery tasks—phase retrieval and blind deconvolution—and match the best known guarantees under Gaussian measurement models and establish new guarantees under heavy-tailed distributions. [ABSTRACT FROM AUTHOR]

Properly defining the center of a set has been a longstanding question in applied mathematics, with implications in numerical geometry, physics, and optimization algorithms. Minkowski centers are one such definition, whose theoretical benefits are numerous and well documented. In this paper, we revisit the advantages of Minkowski centers from a computational, rather than theoretical, perspective. First, we show that Minkowski centers are solutions to a robust optimization problem. Under this lens, we then provide computationally tractable reformulations or approximations for a series of sets, including polyhedra, polyhedral projections, and intersections of ellipsoids. Computationally, we illustrate that Minkowski centers are viable alternatives to other centers, such as Chebyshev or analytic centers, and can speed up the convergence of numerical algorithms like hit-and-run and cutting-plane methods. We hope our work sheds new and practical light on Minkowski centers and exposes their potential benefits as a computational tool. Centers of convex sets are geometric objects that have received extensive attention in the mathematical and optimization literature, both from a theoretical and practical standpoint. For instance, they serve as initialization points for many algorithms such as interior-point, hit-and-run, or cutting-planes methods. First, we observe that computing a Minkowski center of a convex set can be formulated as the solution of a robust optimization problem. As such, we can derive tractable formulations for computing Minkowski centers of polyhedra and convex hulls. Computationally, we illustrate that using Minkowski centers, instead of analytic or Chebyshev centers, improves the convergence of hit-and-run and cutting-plane algorithms. We also provide efficient numerical strategies for computing centers of the projection of polyhedra and of the intersection of two ellipsoids. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2023.2448. [ABSTRACT FROM AUTHOR]

In this article, we build a pair non-differentiable second-order symmetric multiobjective fractional variational programming models with cone constraints, where each objective function component has a support function for a compact convex set. The (C, ρ, θ)-convexity/(C, ρ, θ)-pseudo-convexity/(C, ρ, θ)-quasiconvexity functions are defined and also, constructed concrete numerical examples for existing such type of functions. The duality results are established by using these aforesaid assumptions. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of duality results for optimization problems with vanishing constraints in nonsmooth case. We formulate Wolfe type dual and establish weak, strong, converse, restricted converse and strict converse duality results for mathematical programs with vanishing constraints involving quasidifferentiable functions. Under the assumption of invex and strictly invex functions with respect to a convex compact set. [ABSTRACT FROM AUTHOR]

Let K ⊂ Cd, d ≥ 1 be compact and M(K) be the convex set of Borel probability measures supported in K. We show that various natural subclasses of M(K) associated with regular asymptotic n-th root behavior of corresponding orthonormal polynomials are dense in the standard weak topology under mild assumptions on K. [ABSTRACT FROM AUTHOR]

This is a brief survey of papers of K R Parthasarathy on weak Markov flows and on extreme points of various convex sets appearing in Probability and Quantum Probability. [ABSTRACT FROM AUTHOR]

In this paper, we studied the AA-iterative algorithm for finding fixed points of the class of nonlinear generalized (α, β)-nonexpansive mappings. First, we proved weak convergence and then proved several strong convergence results of the scheme in a ground setting of uniformly convex Banach spaces. We gave a few numerical examples of generalized (α, β)-nonexpansive mappings to illustrate the major outcomes. One example was constructed over a subset of a real line while the other one was on the two dimensional space with a taxicab norm. We considered both these examples in our numerical computations to show that our iterative algorithm was more effective in the rate of convergence corresponding to other fixed point algorithms of the literature. Some 2D and 3D graphs were obtained that supported graphically our results and claims. As applications of our major results, we solved a class of fractional differential equations, 2D Voltera differential equation, and a convex minimization problem. Our findings improved and extended the corresponding results of the current literature [ABSTRACT FROM AUTHOR]

By using a constant step-size, the convergence analysis of the gradient projection method on the sphere is presented for a closed spherically convex set. This algorithm is applied to discuss copositivity of operators with respect to cones. This approach can also be used to analyse solvability of nonlinear cone-complementarity problems. To our best knowledge this is the first numerical method related to the copositivity of operators with respect to the positive semidefinite cone. Numerical results concerning the copositivity of operators are also provided. [ABSTRACT FROM AUTHOR]

The Shapley–Folkman theorem is a statement about the Minkowski sum of (non-convex) sets, expressing the closeness of the Minkowski sum to convexity in a quantitative manner. This paper establishes similar theorems for integrally convex sets, M ♮ -convex sets, and L ♮ -convex sets, which are major classes of discrete convex sets in discrete convex analysis. [ABSTRACT FROM AUTHOR]

We establish new metric characterizations for the norm (respectively, ultraweak) closure of the convex hull of a bounded set in an arbitrary C ⁎ -algebra (respectively, von Neumann algebra), and provide applications of these results to the majorization theory. [ABSTRACT FROM AUTHOR]

In this paper, we provide a structure theorem and various characterizations of degradable strong entanglement breaking maps on separable Hilbert spaces. In the finite-dimensional case, we prove that unital degradable entanglement breaking maps are precisely the C ⁎ -extreme points of the convex set of unital entanglement breaking maps on matrix algebras. Consequently, we get a structure for unital degradable positive partial transpose (PPT) maps. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating generalized relative interior notions for sets in locally convex topological vector spaces with particular attentions to graphs of set-valued mappings and epigraphs of extended-real-valued functions. We introduce, study, and utilize a novel notion of quasi-near convexity of sets that is an infinite-dimensional extension of the widely acknowledged notion of near convexity. Quasi-near convexity is associated with the quasi-relative interior of sets, which is investigated in the paper together with other generalized relative interior notions for sets, not necessarily convex. In this way, we obtain new results on generalized relative interiors for graphs of set-valued mappings in convexity and generalized convexity settings. [ABSTRACT FROM AUTHOR]

Characteristics of partially pseudo-ordered (K-ordered) rings are considered. Properties of the set of all convex directed ideals in pseudo-ordered rings are described. It is shown that convex directed ideals play for the theory of partially pseudo-ordered rings the same role as convex directed subgroups for the theory of partially ordered groups. Necessary and sufficient conditions for a convex directed ideal of an AO-pseudo-ordered ring to be a rectifying ideal are obtained. We show that the set of all rectifying directed ideals of an AO-pseudo-ordered ring form the root system for the lattice of all convex directed ideals of that ring. Properties of regular ideals for partially pseudo-ordered rings are investigated. Some results are proved concerning convex directed ideals of pseudo-lattice pseudo-ordered rings. [ABSTRACT FROM AUTHOR]

Let Ω = Ω0∖Θ¯ ⊂ ℝn, n ≥ 2, where Ω0 and Θ are two open, bounded and convex sets such that Θ¯ ⊂ Ω0 and let β < 0 be a given parameter. We consider the eigenvalue problem for the Laplace operator associated to Ω, with Robin boundary condition on ∂Ω0 and Neumann boundary condition on ∂Θ. In [47] it is proved that the spherical shell is the only maximizer for the first Robin–Neumann eigenvalue in the class of domains Ω with fixed outer perimeter and volume. We establish a quantitative version of the afore-mentioned isoperimetric inequality; the main novelty consists in the introduction of a new type of hybrid asymmetry, that turns out to be the suitable one to treat the different conditions on the outer and internal boundary. Up to our knowledge, in this context, this is the first stability result in which both the outer and the inner boundary are perturbed. [ABSTRACT FROM AUTHOR]

In the presence of Banach spaces, a novel iterative algorithm is presented in this study using the Chatterjea–Suzuki–C (CSC) condition, and the convergence theorems are established. The efficacy of the proposed algorithm is discussed analytically and numerically. We explain the solution of the Caputo fractional differential problem using our main result and then provide the numerical simulation to validate the results. Moreover, we use MATLAB R (2021a) to compare the obtained numerical results using the new iterative algorithm with some efficient existing algorithms. The work seems to contribute to the current advancement of fixed-point approximation iterative techniques in Banach spaces. [ABSTRACT FROM AUTHOR]

Abdessamad Oussarhana and Tijani AmahroqbaLIMATI Laboratory, Poly-disciplinary Faculty, Sultan Moulay Slimane University, Beni Mellal, Morocco; bLAMAI Laboratory, Faculty of Sciences and Techniques, Cadi Ayyad University, Marrakech, Morocco We provide a counterexample to Proposition 2.3 in our paper [Optimization 20, 69(12), pp. 2611–2627] where we said that the convex set A and the element $ x_0 $ x 0 can be properly separated by a closed hyperplane if $ x_0\notin {\rm pri}(A) $ x 0 ∉ pri (A) , with $ {\rm pri}(A) $ pri (A) is the pseudo relative interior of A. In addition, we re-establish all results rely on this proposition, more precisely, [Amahroq and Oussarhan Infinite-dimensional vector optimization and a separation theorem. Optimization. 2020;69(12):2611–2627, Lemma 2.1 (ii)–(iv)], [Amahroq and Oussarhan Infinite-dimensional vector optimization and a separation theorem. Optimization. 2020;69(12):2611–2627, Theorem 3.1] and [Amahroq and Oussarhan Infinite-dimensional vector optimization and a separation theorem. Optimization. 2020;69(12):2611–2627, Corollary 5.1] (see Section 1). A revision to the proof of [Amahroq and Oussarhan Infinite-dimensional vector optimization and a separation theorem. Optimization. 2020;69(12):2611–2627, Theorem 4.1] is also done (see Section 2). We claim that, the rest of the paper (Amahroq and Oussarhan Infinite-dimensional vector optimization and a separation theorem. Optimization. 2020;69(12):2611–2627) is independent of the above incorrect statements. [ABSTRACT FROM AUTHOR]

Stochastic proximal gradient descent (Prox-SGD) is a standard optimization algorithm for solving stochastic composite optimization problems in machine learning. However, the existing convergence rate analysis of Prox-SGD is mostly focused on convergence in expectation. To this end, we provide an almost sure convergence rate analysis of Prox-SGD in two optimization settings: convexity and strong convexity, showing that any implementation of the stochastic algorithm will converge with probability one. Specifically, we show that the average-iterates almost sure convergence rate of the function values for Prox-SGD is arbitrarily close to the optimal rate $ o(1/\sqrt {T}) $ o (1 / T) in the convex setting. For strongly convex objective functions, we obtain the almost sure convergence rate of the iterative sequence, which is also arbitrarily close to the optimal rate $ o(1/T) $ o (1 / T). Last but not least, we establish the more challenging last-iterate convergence rate of the function values on convex problems, which contrasts with existing results on convergence for a weighted average. [ABSTRACT FROM AUTHOR]