General Convex Research Articles

Quantitative homogenization for combustion in random media

We obtain the first quantitative stochastic homogenization result for reaction–diffusion equations, for ignition reactions in dimensions d\le 3 that either have finite ranges of dependence or are close enough to such reactions, and for solutions with initial data that approximate characteristic functions of general convex sets. We show an algebraic rate of convergence of these solutions to their homogenized limits, which are (discontinuous) viscosity solutions of certain related Hamilton–Jacobi equations.

Normal polytopes: Between discrete, continuous, and random

The first three sections of this survey represent an updated and much expanded version of the abstract of my talk at FPSAC'2010: new results are incorporated and several concrete conjectures on the interactions between the three perspectives on normal polytopes in the title are proposed. The last section outlines new challenges in general convex polytopes, motivated by the study of normal polytopes.

Some Certain Fuzzy Aumann Integral Inequalities for Generalized Convexity via Fuzzy Number Valued Mappings

The topic of convex and nonconvex mapping has many applications in engineering and applied mathematics. The Aumann and fuzzy Aumann integrals are the most significant interval and fuzzy operators that allow the classical theory of integrals to be generalized. This paper considers the well-known fuzzy Hermite–Hadamard (HH) type and associated inequalities. With the help of fuzzy Aumann integrals and the newly introduced fuzzy number valued up and down convexity (UD-convexity), we increase this mileage even further. Additionally, with the help of definitions of lower UD-concave (lower UD-concave) and upper UD-convex (concave) fuzzy number valued mappings (FNVMs), we have gathered a sizable collection of both well-known and new extraordinary cases that act as applications of the main conclusions. We also offer a few examples of fuzzy number valued UD-convexity to further demonstrate the validity of the fuzzy inclusion relations presented in this study.

Existence and regularity of solutions of a supersonic-sonic patch arising in axisymmetric relativistic transonic flow with general equation of state

In this article, we prove the existence and regularity of a smooth solution for a supersonic-sonic patch arising in a modified Frankl problem in the study of three-dimensional axisymmetric steady isentropic relativistic transonic flows over a symmetric airfoil. We consider a general convex equation of state which makes this problem complicated as well as interesting in the context of the general theory for transonic flows. Such type of patches appear in many transonic flows over an airfoil and flow near the nozzle throat. Here the main difficulty is the coupling of nonhomogeneous terms due to axisymmetry and the sonic degeneracy for the relativistic flow. However, using the well-received characteristic decompositions of angle variables and a partial hodograph transformation we prove the existence and regularity of solution in the partial hodograph plane first. Further, by using an inverse transformation we construct a smooth solution in the physical plane and discuss the uniform regularity of solution up to the associated sonic curve. Finally, we also discuss the uniform regularity of the sonic curve.

Convex regularization in statistical inverse learning problems

We consider a statistical inverse learning problem, where the task is to estimate a function $ f $ based on noisy point evaluations of $ Af $, where $ A $ is a linear operator. The function $ Af $ is evaluated at i.i.d. random design points $ u_n $, $ n = 1, ..., N $ generated by an unknown general probability distribution. We consider Tikhonov regularization with general convex and $ p $-homogeneous penalty functionals and derive concentration rates of the regularized solution to the ground truth measured in the symmetric Bregman distance induced by the penalty functional. We derive concrete rates for Besov norm penalties and numerically demonstrate the correspondence with the observed rates in the context of X-ray tomography.

Stones From Other Hills Can Polish the Jade: Exploiting Wireless-Powered Cooperative Jamming for Boosting Wireless Information Surveillance

This paper studies information surveillance over wireless-powered suspicious multiuser communications, where multiple suspicious transmitters (STs) first harvest wireless energy from the suspicious power beacon (PB) in phase I and then communicate with the suspicious destination (SD) in phase II over mutually orthogonal channels, and there is a legitimate monitor (M) aiming to overhear the suspicious signals of the STs based on wireless-powered cooperative jamming. Specifically, the jammers first harvest energy from M in phase I and then interfere with the SD in phase II. Considering the fairness issue, M aims to maximize the minimum eavesdropping success probability of these suspicious signals, by jointly optimizing its transmit power in phase I and the jammers’ power allocations in phase II. To solve the problem, first we strictly prove that M should exhaust its maximum power for the energy transfer, even the additional energy can be harvested by the STs to enhance their transmit power and rate. Then, the general successive convex approximation (SCA) technique and one low-complexity solution are respectively proposed to optimize the jammers’ power allocations. Further, the closed-form jamming power allocations are derived in the high signal-to-noise ratio range to reveal some interesting insights. The joint deployments of M and the jammers are also investigated to enhance the eavesdropping performance. Simulation results show the effectiveness of our proposed schemes compared to competitive benchmarks.

Pseudospectral Convex Programming for Free-Floating Space Manipulator Path Planning

To efficiently plan the point-to-point path for a 7-degrees-of-freedom (7-DOF) free-floating space manipulator system, a path planning method based on Legendre pseudospectral convex programming (LPCP) is proposed. First, the non-convex dynamics are approximated by utilizing the first-order Taylor expansion in the vicinity of the initial guess path, which results in a convex system. Next, the linearized dynamics are discretized at Legendre–Gauss–Lobatto collocation points to transcribe the differential equations to a set of equality constraints. To obtain a reliable initial guess trajectory, the auxiliary path planning problem of the 7-DOF space manipulator with a fixed base is initially resolved. Additionally, the penalty function method is introduced to enhance the convergence performance of the LPCP. Finally, simulation results show that the proposed algorithm in this paper can generate the point-to-point path and has higher computational efficiency than the general sequential convex programming method while ensuring optimality.

Complexity and winning strategies of graph convexity games (Brief Announcement)

Accordingly to Duchet (1987), the first paper of convexity on general graphs, in english, is the 1981 paper “Convexity in graphs”. One of its authors, Frank Harary, introduced in 1984 the first graph convexity games, focused on the geodesic convexity, which were investigated in a sequence of five papers that ended in 2003. In this paper, we continue this research line, extend these games to other graph convexities, and obtain winning strategies and complexity results. Among them, we obtain winning strategies for general convex geometries in graphs. We also obtain the first PSPACE-hardness results on convexity games, by proving that the normal play and the misère play of the hull game on the geodesic and the monophonic convexities are PSPACE-complete.

Decentralized Inexact Proximal Gradient Method With Network-Independent Stepsizes for Convex Composite Optimization

This paper proposes a novel CTA (Combine-Then-Adapt)-based decentralized algorithm for solving convex composite optimization problems over undirected and connected networks. The local loss function in these problems contains both smooth and nonsmooth terms. The proposed algorithm uses uncoordinated network-independent constant stepsizes and only needs to approximately solve a sequence of proximal mappings, which is advantageous for solving decentralized composite optimization problems where the proximal mappings of the nonsmooth loss functions may not have analytical solutions. For the general convex case, we prove an O(1/k) convergence rate of the proposed algorithm, which can be improved to o(1/k) if the proximal mappings are solved exactly. Furthermore, with metric subregularity, we establish a linear convergence rate for the proposed algorithm. Numerical experiments demonstrate the efficiency of the algorithm.

Hierarchical Passivity Criterion for Delayed Neural Networks via A General Delay-Product-Type Lyapunov-Krasovskii Functional.

This article is concerned with passivity analysis of neural networks with a time-varying delay. Several techniques in the domain are improved to establish the new passivity criterion with less conservatism. First, a Lyapunov-Krasovskii functional (LKF) is constructed with two general delay-product-type terms which contain any chosen degree of polynomials in time-varying delay. Second, a general convexity lemma without conservatism is developed to address the positive-definiteness of the LKF and the negative-definiteness of its time-derivative. Then, with these improved results, a hierarchical passivity criterion of less conservatism is obtained for neural networks with a time-varying delay, whose size and conservatism vary with the maximal degree of the time-varying delay polynomial in the LKF. It is shown that the conservatism of the passivity criterion does not always reduce as the degree of the time-varying delay polynomial increases. Finally, a numerical example is given to illustrate the proposed criterion and benchmark against the existing results.

Weighted Hermite-Hadamard integral inequalities for general convex functions.

In this article, starting with an equation for weighted integrals, we obtained several extensions of the well-known Hermite-Hadamard inequality. We used generalized weighted integral operators, which contain the Riemann-Liouville and the $ k $-Riemann-Liouville fractional integral operators. The functions for which the operators were considered satisfy various conditions such as the $ h $-convexity, modified $ h $-convexity and $ s $-convexity.

Learner-Private Convex Optimization

Convex optimization with feedback is a framework where a learner relies on iterative queries and feedback to arrive at the minimizer of a convex function. It has gained considerable popularity thanks to its scalability in large-scale optimization and machine learning. The repeated interactions, however, expose the learner to privacy risks from eavesdropping adversaries that observe the submitted queries. In this paper, we study how to optimally obfuscate the learner’s queries in convex optimization with first-order feedback, so that their learned optimal value is provably difficult to estimate for an eavesdropping adversary. We consider two formulations of learner privacy: a Bayesian formulation in which the convex function is drawn randomly, and a maximin formulation in which the function is fixed and the adversary’s probability of error is measured with respect to a minimax criterion. Suppose that the learner wishes to ensure the adversary cannot estimate accurately with probability greater than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1/L$ </tex-math></inline-formula> for some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L &gt; 0$ </tex-math></inline-formula> . Our main results show that the query complexity overhead is additive in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> in the maximin formulation, but multiplicative in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> in the Bayesian formulation. Compared to existing learner-private sequential learning models with binary feedback, our results apply to the significantly richer family of general convex functions with full-gradient feedback. Our proofs rely on tools from the theory of Dirichlet processes, as well as a novel strategy designed for measuring information leakage under a full-gradient oracle.

An analysis of the isoparametric bilinear finite volume element method by applying the Simpson rule to quadrilateral meshes

&lt;abstract&gt;&lt;p&gt;In this work, we construct and study a special isoparametric bilinear finite volume element scheme for solving anisotropic diffusion problems on general convex quadrilateral meshes. The new scheme is obtained by employing the Simpson rule to approximate the line integrals in the classical isoparametric bilinear finite volume element method. By using the cell analysis approach, we suggest a sufficient condition to ensure the coercivity of the new scheme. The sufficient condition has an analytic expression, which only involves the anisotropic diffusion tensor and the geometry of quadrilateral mesh. This yields that for any diffusion tensor and quadrilateral mesh, we can directly judge whether this sufficient condition is satisfied. Specifically, this condition covers the traditional $ h^{1+\gamma} $-parallelogram and some trapezoidal meshes with any full anisotropic diffusion tensor. An optimal $ H^1 $ error estimate of the proposed scheme is also obtained for a quasi-parallelogram mesh. The theoretical results are verified by some numerical experiments.&lt;/p&gt;&lt;/abstract&gt;

О роли множителей Лагранжа и двойственности в некорректных задачах на условный экстремум. К 60-летию метода регуляризации Тихонова

The important role of Lagrange multipliers and duality in the theory of ill-posed problems for a constrained extremum is discussed. The central attention is paid to the problem of stable approximate finding of a normal (minimum in norm) solution of the operator equation of the first kind Az=u, z∈D⊆Z, where A:Z→U is a linear bounded operator, u∈U is a given element, D⊆Z is a convex closed set, Z,U are Hilbert spaces. As is known, this problem is classical for the theory of ill-posed problems. We consider two problems equivalent to it (from the point of view of the simultaneous existence of their unique solutions) for a constrained extremum, the first of which is the problem (CE1) with a functional inequality constraint 〖∥z∥〗^2→"min" , 〖∥Az-u∥〗^2≤0, z∈D, and the second is the problem (CE2) with operator equality constraint 〖∥z∥〗^2→"min" , Az=u, z∈D. First of all, we show that Tikhonov’s regularization method can be naturally interpreted as a method of stable approximation of the exact solution by extremals of the Lagrange functional for problem (CE1) with simultaneous construction of a maximizing sequence of Lagrange multipliers in its dual problem. In this case, the Lagrange multiplier is the reciprocal of the regularization parameter in the Tikhonov method. In other words, the convergence theorem of the Tikhonov regularization method is given the form of a statement in the form of duality with respect to the problem (CE1). Next, we discuss the role of Tikhonov stabilization for general convex problems in solving problems for constrained extremum and a stable method based on Tikhonov stabilization of the problem dual to (CE2) for solving the original operator equation, which can be considered as a regularization method for the Lagrange multiplier rule for the problem (CE2). The paper discusses the features of each of the two above mentioned approaches to the regularization of solving the original operator equation.

Joint Optimization of URLLC Parameters and Beamforming Design for Multi-RIS-Aided MU-MISO URLLC System

This letter focuses on a novel resource allocation design for a multi-reconfigurable intelligent surface (RIS)-aided multi-input and single-output (MISO) ultra-reliable low-latency communication (URLLC) system. In particular, we formulate the sum-rate maximization problem for joint optimization of block-length and packet-error probability for each user, precoder design at the base-station (BS) and passive beamforming at the RIS. To solve it, we provide an alternating optimization framework which involves the decoupling of original problem into the sub-problems of beamforming design and block-length optimization, and later solving them using general convex approximations. Numerical results validate the fast convergence of the proposed solution. Moreover, the proposed solution achieves upto 40% better performance when compared to random resource allocation and each RIS deployment boost the rate performance by 3–5%.

Predefined-Time Leader-Following Formation Control of Nonlinear Multiagent Systems under Switching Topologies

The paper proposes a predefined-time leader-following formation control scheme for nonlinear multi-agent systems with external perturbations under switching topologies. Unlike previous work on predefined-time containment control, we aim to create an isosceles triangle shaped formation instead of being covered by a generic convex hull by having followers follow the leader's trajectories. To ensure convergence along the sliding surface within a predefined-time, we present a nonlinear sliding surface. A terminal sliding mode controller is then used to obtain the desired formation within a predefined-time. The convergence time is independent of both initial values and control parameters. Using, common Lyapunov function, predefined-time stability of closed-loop system is investigated. Simulation examples demonstrate that the developed control protocol is feasible and robust.

Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations

This article develops dual variational formulations for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis and duality theory. The main duality principle is developed as an application to a Ginzburg–Landau-type system in superconductivity in the absence of a magnetic field. In the first section, we develop new general dual convex variational formulations, more specifically, dual formulations with a large region of convexity around the critical points, which are suitable for the non-convex optimization for a large class of models in physics and engineering. Finally, in the last section, we present some numerical results concerning the generalized method of lines applied to a Ginzburg–Landau-type equation.

Avoiding bad steps in Frank-Wolfe variants

The study of Frank-Wolfe (FW) variants is often complicated by the presence of different kinds of “good” and “bad” steps. In this article, we aim to simplify the convergence analysis of specific variants by getting rid of such a distinction between steps, and to improve existing rates by ensuring a non-trivial bound at each iteration. In order to do this, we define the Short Step Chain (SSC) procedure, which skips gradient computations in consecutive short steps until proper conditions are satisfied. This algorithmic tool allows us to give a unified analysis and converge rates in the general smooth non convex setting, as well as a linear convergence rate under a Kurdyka-Łojasiewicz (KL) property. While the KL setting has been widely studied for proximal gradient type methods, to our knowledge, it has never been analyzed before for the Frank-Wolfe variants considered in the paper. An angle condition, ensuring that the directions selected by the methods have the steepest slope possible up to a constant, is used to carry out our analysis. We prove that such a condition is satisfied, when considering minimization problems over a polytope, by the away step Frank-Wolfe (AFW), the pairwise Frank-Wolfe (PFW), and the Frank-Wolfe method with in face directions (FDFW).

Nonasymptotic Convergence Rates for the Plug-in Estimation of Risk Measures

Let ρ be a general law-invariant convex risk measure, for instance, the average value at risk, and let X be a financial loss, that is, a real random variable. In practice, either the true distribution μ of X is unknown, or the numerical computation of [Formula: see text] is not possible. In both cases, either relying on historical data or using a Monte Carlo approach, one can resort to an independent and identically distributed sample of μ to approximate [Formula: see text] by the finite sample estimator [Formula: see text] (μNdenotes the empirical measure of μ). In this article, we investigate convergence rates of [Formula: see text] to [Formula: see text]. We provide nonasymptotic convergence rates for both the deviation probability and the expectation of the estimation error. The sharpness of these convergence rates is analyzed. Our framework further allows for hedging, and the convergence rates we obtain depend on neither the dimension of the underlying assets nor the number of options available for trading.Funding: Daniel Bartl is grateful for financial support through the Vienna Science and Technology Fund [Grant MA16-021] and the Austrian Science Fund [Grants ESP-31 and P34743]. Ludovic Tangpi is supported by the National Science Foundation [Grant DMS-2005832] and CAREER award [Grant DMS-2143861].

Fiber Convex Bodies

In this paper we study the fiber bodies, that is the extension of the notion of fiber polytopes for more general convex bodies. After giving an overview of the properties of the fiber bodies, we focus on three particular classes of convex bodies. First we describe the strict convexity of the fiber bodies of the so called puffed polytopes. Then we provide an explicit equation for the support function of the fiber bodies of some smooth convex bodies. Finally we give a formula that allows to compute the fiber bodies of a zonoid with a particular focus on certain zonoids called discotopes. Throughout the paper we illustrate our results with detailed examples.