Sufficient Conditions For Convexity Research Articles

Polynomial convexity with degree bounds

We introduce different notions of polynomial convexity with bounds on degrees of polynomials in C n . We provide some examples in higher dimensions and show necessary and sufficient conditions for polynomial convexity with degree bounds for certain sets of points in C and for certain arcs in the unit circle.

Starlikeness, Convexity, Close-to-Convexity, and Quasi-Convexity for Functions with Fixed Initial Coefficients

In this paper, we employ the theory of differential subordination to establish a theorem that delineates certain sufficient conditions for starlikeness, convexity, close-to-convexity, and quasi-convexity in relation to functions with fixed initial coefficients. Furthermore, we introduce some results derived from these conditions. Building upon this framework, we derive an extension of Nunokawa’s lemma for analytic functions with fixed initial coefficients.

The convexity condition of density-functional theory.

It has long been postulated that within density-functional theory (DFT), the total energy of a finite electronic system is convex with respect to electron count so that 2Ev[N0] ≤ Ev[N0 - 1] + Ev[N0 + 1]. Using the infinite-separation-limit technique, this Communication proves the convexity condition for any formulation of DFT that is (1) exact for all v-representable densities, (2) size-consistent, and (3) translationally invariant. An analogous result is also proven for one-body reduced density matrix functional theory. While there are known DFT formulations in which the ground state is not always accessible, indicating that convexity does not hold in such cases, this proof, nonetheless, confirms a stringent constraint on the exact exchange-correlation functional. We also provide sufficient conditions for convexity in approximate DFT, which could aid in the development of density-functional approximations. This result lifts a standing assumption in the proof of the piecewise linearity condition with respect to electron count, which has proven central to understanding the Kohn-Sham bandgap and the exchange-correlation derivative discontinuity of DFT.

Optimal transport and variational Bayesian inference

We apply Lott–Sturm–Villani notion of Ricci curvature as well as McCann–Brenier theorem on optimal transport investigating mean-field variational inference (MFVI). In the continuous case where the probability rules are defined on Riemannian manifolds we obtain sufficient conditions for convexity of the corresponding variational problem. It can be shown that in the presence of symmetry the convexity can provide the preservation of mode. We then consider factorization into a mixed continuous-discrete case. By rescaling the Riemannian structure of the continuous component we have derived sufficient conditions under which the convexity of Kullback-Leibler functional is established. We apply the latter theorem to Bayesian Gaussian Mixture Model (GMM) and we introduce a new class of these models which have the advantage of being convex. It is shown that the MAP for the corrected GMM remains a consistent estimator. Additionally thanks to convexity we are able to prove the mode of MFVI for the new model in asymptotic regimes converges to the right solution. We do not know any other rigorous result for MFVI in statistical learning.

Event-Triggered Neural Network Control for LTI Systems

This paper addresses the problem of event-triggered control (ETC) of discrete-time linear time-invariant (LTI) systems stabilized by neural network controllers. The event-triggering mechanisms (ETMs) is proposed to update only a portion of the layers required to maintain stability and satisfactory performance of the feedback system, thus reducing the computational cost associated with the evaluation of the neural network. Sufficient convex conditions in the form of linear matrix inequalities (LMIs) are provided to compute the triggering parameters and characterize an estimate of the domain of attraction for the feedback system. The formulation is based on the use of Lyapunov theory and a set of generalized sector constraints that deal with the nonlinear activation functions, represented in this case by the saturation function. Optimization procedures are also formulated to effectively reduce the amount of computation in the neural network. An example borrowed from the literature is used to illustrate the effectiveness of the proposal.

Linearly-Involved Moreau-Enhanced-Over-Subspace Model: Debiased Sparse Modeling and Stable Outlier-Robust Regression

We present an efficient mathematical framework based on the linearly-involved Moreau-enhanced-over-subspace (LiMES) model. Two concrete applications are considered: sparse modeling and robust regression. The popular minimax concave (MC) penalty for sparse modeling subtracts, from the $\ell_1$ norm, its Moreau envelope, inducing nearly unbiased estimates and thus yielding remarkable performance enhancements. To extend it to underdetermined linear systems, we propose the projective minimax concave penalty using the projection onto the input subspace, where the Moreau-enhancement effect is restricted to the subspace for preserving the overall convexity. We also present a novel concept of stable outlier-robust regression which distinguishes noise and outlier explicitly. The LiMES model encompasses those two specific examples as well as two other applications: stable principal component pursuit and robust classification. The LiMES function involved in the model is an ``additively nonseparable'' weakly convex function but is defined with the Moreau envelope returning the minimum of a ``separable'' convex function. This mixed nature of separability and nonseparability allows an application of the LiMES model to the underdetermined case with an efficient algorithmic implementation. Two linear/affine operators play key roles in the model: one corresponds to the projection mentioned above and the other takes care of robust regression/classification. A necessary and sufficient condition for convexity of the smooth part of the objective function is studied. Numerical examples show the efficacy of LiMES in applications to sparse modeling and robust regression.

Sparse Stable Outlier-Robust Signal Recovery Under Gaussian Noise

This paper presents a novel framework for sparse robust signal recovery integrating the sparse recovery using the minimax concave (MC) penalty and robust regression called sparse outlier-robust regression (SORR) using the MC loss. While the proposed approach is highly robust against huge outliers, the sparseness of estimates can be controlled by taking into consideration a tradeoff between sparseness and robustness. To accommodate the prior information about additive Gaussian noise and outliers, an auxiliary vector to model the noise is introduced. The remarkable robustness and stability come from the use of the MC loss and the squared <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> penalty of the noise vector, respectively. In addition, the simultaneous use of the MC and squared <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> penalties of the coefficient vector leads to a certain remarkable grouping effect. The necessary and sufficient conditions for convexity of the smooth part of the cost are derived under a certain nonempty-interior assumption via the product space formulation using the linearly-involved Moreau-enhanced-over-subspace (LiMES) framework. The efficacy of the proposed method is demonstrated by simulations in its application to speech denoising under highly noisy environments as well as to toy problems.

Geometric properties of normalized Rabotnov function

In the present paper, our aim is to study geometric properties of normalized Rabotnov functions. For this purpose, we determined sufficient conditions for univalency, close-to-convexity, convexity and starlikeness of the normalized Rabotnov functions in the open unit disk.

Hadamard Product of Certain Multivalent Analytic Functions with Positive Real Parts

This paper aims to provide sufficient conditions for starlikeness and convexity of Hadamard product (convolution) of certain multivalent analytic functions with positive real parts. Moreover, the starlikeness conditions for a certain integral operator and other convolution results are also considered.

Generalized Bregman Envelopes and Proximity Operators

Every maximally monotone operator can be associated with a family of convex functions, called the Fitzpatrick family or family of representative functions. Surprisingly, in 2017, Burachik and Martínez-Legaz showed that the well-known Bregman distance is a particular case of a general family of distances, each one induced by a specific maximally monotone operator and a specific choice of one of its representative functions. For the family of generalized Bregman distances, sufficient conditions for convexity, coercivity, and supercoercivity have recently been furnished. Motivated by these advances, we introduce in the present paper the generalized left and right envelopes and proximity operators, and we provide asymptotic results for parameters. Certain results extend readily from the more specific Bregman context, while others only extend for certain generalized cases. To illustrate, we construct examples from the Bregman generalizing case, together with the natural “extreme” cases that highlight the importance of which generalized Bregman distance is chosen.

On the convexity of the reachable set with respect to a part of coordinates at small time intervals

We investigate the convexity of the reachable sets for some of the coordinates of nonlinear systems with integral constraints on the control at small time intervals. We have proved sufficient convexity conditions in the form of constraints on the asymptotics of the eigenvalues of the Gramian of the controllability of a linearized system for some of the coordinates. There are two nonlinear third-order systems under study as examples. The system linearized along a trajectory generated by zero control is uncontrollable, and the system in the other example is completely controllable. We investigate the sufficient conditions for convexity of projection of reachable sets. Numerical modeling has been carried out, demonstrating the non-convexity of some projections even for small time intervals.

When is the rate function of a random vector strictly convex?

We give a necessary and sufficient condition for strict convexity of the rate function of a random vector in Rd. This condition is always satisfied when the random vector has finite Laplace transform. We also completely describe the effective domain of the rate function under a weaker condition.

Convex Nonparametric Formulation for Identification of Gradient Flows

We develop a nonparametric identification method for nonlinear gradient-flow dynamics. In these systems, the vector field is the gradient field of a potential energy function. This fundamental fact about the dynamics is a structural prior knowledge and a constraint in the proposed identification method. While the nature of the identification problem is an estimation in the space of functions, we derive an equivalent finite dimensional formulation, which is a convex optimization in the form of a quadratic program. This provides scalability and the opportunity for utilizing recently developed large-scale optimization solvers. The central idea is representing the energy function as a difference of two convex functions and learning these convex functions jointly. Based on necessary and sufficient conditions for function convexity, the identification problem is formulated, and then, the existence, uniqueness and smoothness of the solution is addressed. We also illustrate and evaluate the method numerically with two demonstrative examples.

Relationships between convexity and generalized convexity of comparable fuzzy functions

In this paper, we show some relations between the convexity and generalized convexity such as pseudo-convexity, strongly quasi-convexity of fuzzy mappings of several multivariables by using a partial order relation. Also we established a necessary and sufficient condition for convexity of fuzzy mapping. Finally we showed a necessary condition for global minimum of a fuzzy mapping.

A Sufficient Convexity Condition for Parametric B&amp;#233;zier Surface over Rectangle

Surface convexity is a key issue in computer aided geometric design, which is widely applied in geometric modeling field, such as physical models, industrial design, automatic manufacturing, etc. In this paper, a sufficient convexity condition of the parametric Bezier surface over rectangles is proposed, which is firstly considered as a sufficient convexity condition for the Bezier control grid. The condition is proved by De Casteljau surface subdivision arithmetic, in which the recursive expressions elaborate that the control grid eventually converges to the surface. At last, two examples for the modeling of interpolation-type surface are discussed, one of which is a general surface and the other is a degenerate surface.

Geometric Properties of Lommel Functions of the First Kind

In the present paper, we find sufficient conditions for starlikeness and convexity of normalized Lommel functions of the first kind using the admissible function methods. Additionally, we investigate some inclusion relationships for various classes associated with the Lommel functions. The functions belonging to these classes are related to the starlike functions, convex functions, close-to-convex functions and quasi-convex functions.

Certain sufficient conditions for starlikeness and convexity using a multiplier transformation

In the present paper, we study a differential subordination involving a multiplier transformation. Selecting different dominants to our main result, we obtain certain sufficient conditions for starlikeness and convexity of analytic functions. In particular, we obtain the sufficient conditions for parabolic starlikeness and uniform convexity. Some known results appear as particular cases of our main result.

Partial Sampled-Data State Feedback Control of Markov Jump Linear Systems

Abstract This paper aims at designing a partial sampled-data state feedback control law for Markov jump linear systems (MJLS). The interesting feature of the control structure is that only the state variable is sampled, while the stochastic parameter that defines the Markov mode of the system used for control purposes is free to change at any time between samples. The main goal is to provide sufficient convex conditions for the existence of a solution for this class of control design problems in the context of H∞ and H2 performances, which are expressed through Differential Linear Matrix Inequalities (DLMI). The proposed method is implemented using LMIs facilities and provides a minimum guaranteed cost control in only one shot. An example is solved for illustration and comparison.

The perpetual American put option in jump-to-default models

We study the perpetual American put option in a general jump-to-default model, deriving an explicit expression for the price of the option. We find that in some cases the optimal stopping boundary vanishes and thus it is not optimal to exercise the option before default occurs. Precise conditions for when this situation arises are given. Furthermore we present a necessary and sufficient condition for convexity of the option price, and also show that a nonincreasing intensity is sufficient, but not necessary, to have convexity. From this we also get conditions for when option prices are monotone in the model parameters.

Partial stabilizability and hidden convexity of indefinite LQ problem

Generalization of linear system stability theory and LQ control theory are presented. It is shown that the partial stabilizability problem is equivalent to a Linear Matrix Inequality (LMI). Also, the set of all initial conditions for which the system is stabilizable by an open-loop control (the stabilizability subspace) is characterized in terms of a semi-definite programming (SDP). Next, we give a complete theory for an infinite-time horizon Linear Quadratic (LQ) problem with possibly indefinite weighting matrices for the state and control. Necessary and sufficient convex conditions are given for well-posedness as well as attainability of the proposed (LQ) problem. There is no prior assumption of complete stabilizability condition as well as no assumption on the quadratic cost. A generalized algebraic Riccati equation is introduced and it is shown that it provides all possible optimal controls. Moreover, we show that the solvability of the proposed indefinite LQ problem is equivalent to the solvability of a specific SDP problem.