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In this paper, we use Aleksandrov's reflection principle to prove symmetry of solutions to F (∇ S n 2 u + u I) = f (u , u 2 + | ∇ S n u | 2 ) , where u is the support function of a convex body, and F is a function of principal radii. As a corollary, alongside [Ivaki, arXiv:2307.06252 ], we provide an alternative proof of the uniqueness of solution to the isotropic Gaussian-Minkowski problem. [ABSTRACT FROM AUTHOR]

The study of greedy approximation in the context of convex optimization is becoming a promising research direction as greedy algorithms are actively being employed to construct sparse minimizers for convex functions with respect to given sets of elements. In this paper we propose a unified way of analyzing a certain kind of greedy-type algorithms for the minimization of convex functions on Banach spaces. Specifically, we define the class of Weak Biorthogonal Greedy Algorithms for convex optimization that contains a wide range of greedy algorithms. We analyze the introduced class of algorithms and establish the properties of convergence, rate of convergence, and numerical stability, which is understood in the sense that the steps of the algorithm are allowed to be performed not precisely but with controlled computational inaccuracies. We show that the following well-known algorithms for convex optimization — the Weak Chebyshev Greedy Algorithm (co) and the Weak Greedy Algorithm with Free Relaxation (co) — belong to this class, and introduce a new algorithm — the Rescaled Weak Relaxed Greedy Algorithm (co). Presented numerical experiments demonstrate the practical performance of the aforementioned greedy algorithms in the setting of convex minimization as compared to optimization with regularization, which is the conventional approach of constructing sparse minimizers. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of structured fractional minimization problems where the numerator part of the objective is the sum of a convex function and a Lipschitz differentiable (possibly) nonconvex function, while the denominator part is a convex function. By exploiting the structure of the problem, we propose a first-order algorithm, namely, a proximal-gradient-subgradient algorithm with backtracked extrapolation (PGSA_BE) for solving this type of optimization problem. It is worth pointing out that there are a few differences between our backtracked extrapolation and other popular extrapolations used in convex and nonconvex optimization. One of such differences is as follows: if the new iterate obtained from the extrapolated iteration satisfies a backtracking condition, then this new iterate will be replaced by the one generated from the non-extrapolated iteration. We show that any accumulation point of the sequence generated by PGSA_BE is a critical point of the problem regarded. In addition, by assuming that some auxiliary functions satisfy the Kurdyka-Łojasiewicz property, we are able to establish global convergence of the entire sequence, in the case where the denominator is locally Lipschitz differentiable, or its conjugate satisfies the calmness condition. Finally, we present some preliminary numerical results to illustrate the efficiency of PGSA_BE. [ABSTRACT FROM AUTHOR]

The C 1 , 1 estimate of the Dirichlet problem for degenerate k -Hessian equations with non-homogenous boundary conditions is an open problem, if the right hand side function f is only assumed to satisfy f 1 / (k − 1) ∈ C 1 , 1. In this paper, we solve this problem for convex solutions defined in the strictly convex bounded domain. [ABSTRACT FROM AUTHOR]

In mathematical modelling, the data and solutions are represented as measurable functions and their quality is oftentimes captured by the membership to a certain function space. One of the core questions for an analysis of a model is the mutual relationship between the data and solution quality. The optimality of the obtained results deserves a special focus. It requires a careful choice of families of function spaces balancing between their expressivity, i.e. the ability to capture fine properties of the model, and their accessibility, i.e. its technical difficulty for practical use. This paper presents a unified and general approach to optimality problems in Orlicz spaces. Orlicz spaces are parametrized by a single convex function and neatly balance the expressivity and accessibility. We prove a general principle that yields an easily verifiable necessary and sufficient condition for the existence or the non-existence of an optimal Orlicz space in various tasks. We demonstrate its use in specific problems, including the continuity of Sobolev embeddings and boundedness of integral operators such as the Hardy–Littlewood maximal operator and the Laplace transform. [ABSTRACT FROM AUTHOR]

This paper is concerned with various fine properties of the functional D (A) ≐ ∫ T n det 1 n − 1 ⁡ (A (x)) d x introduced in [34]. This functional is defined on X p , which is the cone of matrix fields A ∈ L p (T n ; Sym + (n)) with div (A) a bounded measure. We start by correcting a mistake we noted in our [13, Corollary 7] , which concerns the upper semicontinuity of D (A) in X p. We give a proof of a refined correct statement, and we will use it to study the behaviour of D (A) when A ∈ X n n − 1 , which is the critical integrability for D (A). One of our main results gives an explicit bound of the measure generated by D (A k) for a sequence of such matrix fields { A k } k. In particular it allows us to characterize the upper semicontinuity of D (A) in the case A ∈ X n n − 1 in terms of the measure generated by the variation of { div A k } k. We show by explicit example that this characterization fails in X p if p < n n − 1. As a by-product of our characterization we also recover and generalize a result of P.-L. Lions [26,27] on the lack of compactness in the study of Sobolev embeddings. Furthermore, in analogy with Monge-Ampère theory, we give sufficient conditions under which det 1 n − 1 ⁡ (A) ∈ H 1 (T n) when A ∈ X n n − 1 , generalising the celebrated result of S. Müller [30] when A = cof D 2 φ , for a convex function φ. [ABSTRACT FROM AUTHOR]

The lighting situation in which a picture was taken has an impact on its color. Illuminant estimation is crucial in computer vision because the colors of objects vary as illumination changes. For this reason, numerous methods for estimating the illuminant have been suggested. In this paper, we suggest a novel statistic-based method for estimating single and multiple illuminants using convex functions. In this respect, convex functions are used in the two subsequent steps of normalization and weight creation. After using weighted K-means to segment the picture, each segment's associated illuminations are determined. The illumination map for the input image is estimated as a final stage. In this study, we also analyze the effect of convexity on color constancy algorithms and present proofs for the convexity of some statistic-based algorithms. Four different single and multi-illuminant datasets have been used to evaluate the proposed algorithm in terms of two evaluation metrics; recovery and reproduction angular error. We believe that the proposed method could be considered one of the statistical state-of-the-art algorithms. In addition, it has competitive results when compared to most learning-based and deep-learning methods. Further advantages of the proposed algorithm include its simplicity of implementation and low execution time. [Display omitted] • A new statistic-based method with easy implementation and no learning required. • A general statistic-based method for estimating single and multiple illuminations. • Presenting an efficient method that is one of the statistical state-of-the-art methods. • Estimating the local and global illuminations using convex functions. • Algorithm stability against changing the number of light sources and databases. [ABSTRACT FROM AUTHOR]

Alternative Tuned Mass Damper (TMD) configurations have been proposed in the literature for performance improvement and maintenance, mainly modifying their restoring force or dissipation mechanisms. In this context, Folded-Pendulum TMDs (FPTMDs) are appealing for tuning low-frequency structures because they are compact and easy to retune while avoiding sliding mechanisms. In turn, Friction Dampers (FDs) are equally attractive due to their minimal maintenance and easy desired capacity adjustment. Nevertheless, despite recent advances in integrating FDs with PTMDs, a comprehensive literature survey reveals that Reliability-Based Design Optimization (RBDO) studies for PTMDs or Folded-PTMDs with FDs cannot be found to the best of the authors' knowledge. Hence, the present paper aims at performing an original RBDO of single and multiple Friction-Folded-PTMDs in buildings subjected to stochastic excitation. Unlike the existing optimization studies in this field, a complete nonlinear description (dry friction plus large rotations) is employed because the considered applications involve high-intensity excitations, like wind and seismic loading. Finally, scenarios with multiple absorbers are included due to their associated performance and constructional advantages. Optimal design of multiple Folded-PTMDs with FDs has not yet been addressed, as they increase the optimization problem complexity leading to multimodal and convex objective functions. Accordingly, an active-learning Kriging-based Efficient Global Optimization (EGO) procedure is used, which allows for finding the optimum solutions with only a few objective function evaluations. The application cases to two distinct buildings subject to wind and seismic loadings show that using Friction-FPTMDs allows for obtaining a control strategy with an appropriate performance while ensuring practical advantages over classical TMDs. [ABSTRACT FROM AUTHOR]

• The compression sampling and reconstruction of road roughness signal were realized. • The proposed ISA-SPA reconstruction method can adaptively match sparsity. • SA-GSM-SPA reconstruction method matched sparsity accurately and consumed less time. • The method presented in the paper was validated on hard pavement and soft pavement. • Sparsity matching equaled to extremal process of convex function approximately. To significantly reduce the storage space of collected road roughness signals and improve the collection rate, the work investigates the compressive sampling and reconstruction of road roughness signals based on compressive sensing theory. Moreover, to overcome the limitations of the classical signal reconstruction method in the case of unknown sparsity, two sparsity adaptive compressive signal reconstruction methods namely those based on the improved simulated annealing (I-SA) algorithm and the golden section method (GSM) are respectively proposed and compared. Both simulated and measured road roughness signals are used to verify the validity of the novel reconstruction methods and the feasibility of road roughness compression and collection. The research results show that the proposed ISA-SPA method (the sparsity adaptive reconstruction method based on the I-SA) completes the sparsity matching optimization with high reconstruction precision (R 2 = 0.9884), but has a high time consumption (about 16.09 s). Moreover, the proposed SA-GSM-SPA method (the sparsity adaptive reconstruction method based on the GSM) has a fast rate of calculation while inheriting the good sparsity estimation result and high reconstruction precision of the ISA-SPA method. [ABSTRACT FROM AUTHOR]

The classic Riesz representation theorem characterizes all linear and increasing functionals on the space C c (X) of continuous compactly supported functions. A geometric version of this result, which characterizes all linear increasing functionals on the set of convex bodies in R n , was essentially known to Alexandrov. This was used by Alexandrov to prove the existence of mixed area measures in convex geometry. In this paper we characterize linear and increasing functionals on the class of log-concave functions on R n. Here "linear" means linear with respect to the natural addition on log-concave functions which is the sup-convolution. Equivalently, we characterize pointwise-linear and increasing functionals on the class of convex functions. For some choices of the exact class of functions we prove that there are no non-trivial such functionals. For another choice we obtain the expected analogue of the result for convex bodies. And most interestingly, for yet another choice we find a new unexpected family of such functionals. Finally, we explain the connection between our results and recent work done in convex geometry regarding the surface area measure of a log-concave functions. An application of our results in this direction is also given. [ABSTRACT FROM AUTHOR]

Murota (1998) and Murota and Shioura (1999) introduced concepts of M-convex function and M♮-convex function as discrete convex functions, which are generalizations of valuated matroids due to Dress and Wenzel (1992). In the present paper we consider a new operation defined by a convolution of sections of an M♮-convex function that transforms the given M♮-convex function to an M-convex function, which we call a compression of an M♮-convex function. For the class of valuated generalized matroids, which are special M♮-convex functions, the compression induces a valuated permutohedron together with a decomposition of the valuated generalized matroid into flag-matroid strips , each corresponding to a maximal linearity domain of the induced valuated permutohedron. We examine the details of the structure of flag-matroid strips and the induced valuated permutohedron by means of discrete convex analysis of Murota. [ABSTRACT FROM AUTHOR]

In recent years, quaternion matrix completion (QMC) based on low-rank regularization has been gradually used in image processing. Unlike low-rank matrix completion (LRMC) which handles RGB images by recovering each color channel separately, QMC models retain the connection of three channels and process them as a whole. Most of the existing quaternion-based methods formulate low-rank QMC (LRQMC) as a quaternion nuclear norm (a convex relaxation of the rank) minimization problem. The main limitation of these approaches is that they minimize the singular values simultaneously such that cannot approximate low-rank attributes efficiently. To achieve a more accurate low-rank approximation, we introduce a quaternion truncated nuclear norm (QTNN) for LRQMC and utilize the alternating direction method of multipliers (ADMM) to get the optimization in this paper. Further, we propose weights to the residual error quaternion matrix during the update process for accelerating the convergence of the QTNN method with admissible performance. The weighted method utilizes a concise gradient descent strategy which has a theoretical guarantee in optimization. The effectiveness of our method is illustrated by experiments on real visual data sets. • A novel quaternion truncated nuclear norm is proposed for low-rank quaternion completion problem. • An efficient alternating direction method of multipliers is developed to solve the proposed method. • Further, we add weighted matrices during the optimization, and a concise gradient descent strategy is developed with a theoretical guarantee. [ABSTRACT FROM AUTHOR]