Continuous Convex Function Research Articles

On locally Lipschitz convex functions defined on subsets with empty interior of locally convex spaces

The generic Fréchet and/or Gateaux differentiability of a continuous convex function on an open convex subset of a Banach space is well studied and has important theoretical implications. An example of Rainwater [Yet more on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(3):773–778] shows that similar results are not true when the openness of the domain is not ensured, but such properties can be obtained if the respective convex function is assumed to be locally Lipschitz, as shown by Verona [More on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(1):137–140]; Rainwater [Yet more on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(3):773–778]; Noll [Generic Gâteaux-differentiability of convex functions on small sets. J Math Anal Appl. 1990;147(2):531–544] and others. It is our aim to extend such results for convex functions defined on (not necessarily open) convex subsets of locally convex spaces. In the same framework we extend Gale's duality theorem (1967), as well as a result of H. Ergin & T. Sarver on the unique dual representation of a convex function (2010). Moreover, a detailed study of the Rainwater example is done.

New Auxiliary Principle Technique for General Harmonic Directional Variational Inequalities

This paper explores the utilisation of harmonic variational inequalities to establish the minimum value among two locally Lipschitz continuous harmonic convex functions. This investigation introduces novel classes of harmonic directed variational inequalities, particularly focusing on scenarios like harmonic complementarity and related optimization challenges. The study proposes and analyses various inertial iterative strategies for addressing harmonic directed variational inequalities through the auxiliary principle technique. It examines convergence criteria under specific weak conditions, emphasising the simplicity of the approach compared to other methodologies. The findings presented herein have broad applicability in the context of harmonic variational inequalities and optimization problems, though they are limited to theoretical exploration. Further research is required to implement these strategies numerically.

Smoothing algorithms for nonsmooth optimization over the Stiefel manifold with applications to the graph Fourier basis problem

In this paper, we consider a class of nonsmooth and nonconvex optimization problems over the Stiefel manifold where the objective function is the summation of a nonconvex smooth function and a nonsmooth Lipschitz continuous convex function composed with a linear mapping. Besides, we are interested in its application to the graph Fourier basis problem. We propose three numerical algorithms for solving this problem, by combining smoothing methods and some existing algorithms for smooth optimization over the Stiefel manifold. In particular, we approximate the aforementioned nonsmooth convex function by its Moreau envelope in our smoothing methods, and prove that the Moreau envelope has many favorable properties. Thanks to this and the scheme for updating the smoothing parameter, we show that any accumulation point of the solution sequence generated by the proposed algorithms is a stationary point of the original optimization problem. Numerical experiments on building graph Fourier basis are conducted to demonstrate the efficiency of the proposed algorithms.

Parameterized Convex Universal Approximators for Decision-Making Problems.

Parameterized max-affine (PMA) and parameterized log-sum-exp (PLSE) networks are proposed for general decision-making problems. The proposed approximators generalize existing convex approximators, namely max-affine (MA) and log-sum-exp (LSE) networks, by considering function arguments of condition and decision variables and replacing the network parameters of MA and LSE networks with continuous functions with respect to the condition variable. The universal approximation theorem (UAT) of PMA and PLSE is proved, which implies that PMA and PLSE are shape-preserving universal approximators for parameterized convex continuous functions. Practical guidelines for incorporating deep neural networks within PMA and PLSE networks are provided. A numerical simulation is performed to demonstrate the performance of the proposed approximators. The simulation results support that PLSE outperforms other existing approximators in terms of a minimizer and optimal value errors with scalable and efficient computation for high-dimensional cases.

Convex Optimization Problems on Differentiable Sets

Given a closed convex set in a Banach space , motivated by the continuity and Fréchet differentiability of introduced, respectively, in [D. Gale and V. Klee, Math. Scand., 7 (1959), pp. 379–391] and [X. Y. Zheng, SIAM J. Optim., 30 (2020), pp. 490–512], this paper considers the -differentiability, subdifferentiability, and Gâteaux differentiability of . Using the technique of variational analysis, it is proved that is -differentiable (resp., subdifferentiable or Gâteaux differentiable) if and only if for every continuous convex function with the corresponding constrained optimization problem is 2/ -order-well-posed solvable (resp., generalized well-posed solvable or weak well-posed solvable). It is also proved that if the conjugate function of a continuous convex function on is -differentiable on , then for every closed convex set in with the corresponding optimization problem is -order-well-posed solvable. As a byproduct, every constrained convex optimization problem with a strongly convex quadratic objective function is proved to be globally second-order-well-posed solvable. Our main results are new even in the case of finite dimensional spaces.

Regularly abstract convex functions with respect to the set of Lipschitz continuous concave functions

Given a set of functions defined on a set X, á function is called abstract -convex if it is the upper envelope of its -minorants, i.e. such its minorants which belong to the set ; and f is called regularly abstract -convex if it is the upper envelope of its maximal (with respect to the pointwise ordering) -minorants. In the paper we first present the basic notions of (regular) -convexity for the case when is an abstract set of functions. For this abstract case a general sufficient condition based on Zorn's lemma for a -convex function to be regularly -convex is formulated. The goal of the paper is to study the particular class of regularly -convex functions, when is the set of real-valued Lipschitz continuous classically concave functions defined on a real normed space X. For an extended-real-valued function to be -convex it is necessary and sufficient that f be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each -convex function is regularly -convex as well. We focus on -subdifferentiability of functions at a given point. We prove that the set of points at which an -convex function is -subdifferentiable is dense in its effective domain. This result extends the well-known classical Brøndsted-Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the more wide class of lower semicontinuous functions. Using the subset of the set consisting of such Lipschitz continuous concave functions that vanish at the origin we introduce the notions of -subgradient and -subdifferential of a function at a point which generalize the corresponding notions of the classical convex analysis. Some properties and simple calculus rules for -subdifferentials as well as -subdifferential conditions for global extremum points are established. Symmetric notions of abstract -concavity and -superdifferentiability of functions where is the set of Lipschitz continuous convex functions are also considered.

Fast optimization via inertial dynamics with closed-loop damping

In a real Hilbert space \mathcal{H} , in order to develop fast optimization methods, we analyze the asymptotic behavior, as time t tends to infinity, of a large class of autonomous dissipative inertial continuous dynamics. The function f: \mathcal{H} \to \mathbb{R} to be minimized (not necessarily convex) enters the dynamic via its gradient, which is assumed to be Lipschitz continuous on bounded subsets of \mathcal{H} . This results in autonomous dynamical systems with nonlinear damping and nonlinear driving force. We first consider the case where the damping term \partial \phi (\dot{x}(t)) acts as closed-loop control of the velocity. The damping potential \phi : \mathcal{H} \to \mathbb{R}_+ is a convex continuous function which reaches its minimum at the origin. We show the existence and uniqueness of a global solution to the associated Cauchy problem. We analyze the asymptotic convergence and the convergence rates of the trajectories generated by this system. To do this, we use techniques from optimization, control theory, and PDEs: Lyapunov analysis based on the decreasing property of an energy-like function, quasi-gradient and Kurdyka–Łojasiewicz theory, and monotone operator theory for wave-like equations. Convergence rates are obtained based on the geometric properties of the data f and \phi . We put forward minimal hypotheses on the damping potential \phi guaranteeing the convergence of trajectories, thus showing the dividing line between strong and weak damping. When f is strongly convex, we give general conditions on the damping potential \phi which provide exponential convergence rates. Then, we extend the results to the case where additional Hessian-driven damping enters the dynamic, which reduces the oscillations. Finally, we consider a new inertial system where damping jointly involves the velocity \dot{x}(t) and the gradient \nabla f(x(t)) . This study naturally leads to similar results for proximal-gradient algorithms obtained by temporal discretization; some of them are studied in the article. In addition to its original results, this work surveys numerous works devoted to the interaction between damped inertial continuous dynamics and numerical optimization algorithms, with an emphasis on autonomous systems, adaptive procedures, and convergence rates.

Approximation and Gâteaux differentiability of convex function in Banach spaces

AbstractThis paper shows that if f is a convex continuous function on a Gâteaux differentiability space X, then for any separable closed subspace E of X, there exists a sequence of continuous convex functions such that (1) on X; (2) is Gâteaux differentiable at all points of a dense open subset of X; (3) on E. Moreover, if X is separable, then there exists a continuous convex function sequence such that (1) and (2) are true and on X.

Single-machine slack due-window assignment scheduling with multiple maintenance activities and position-and-resource-dependent processing times

This paper studies the slack due-window assignment method for single-machine scheduling problems with position-and resource-dependent processing times, for which the actual processing time is a continuous bivariate non-increasing convex function involving position and resource allocation. Only the slack due-window assignment method will be used. Two linear combination problems concerning position-dependent effects and resource allocation will be studied, in which the machine requires multiple maintenance activities. We present the optimal properties or design optimal algorithms to minimise the combination of the makespan, the total completion time and the total absolute deviation of completion time and to minimise the combination of the total costs of tardiness, the slack due date, the earliness cost, and the resource amount. Finally, we also consider the bi-criteria scheduling problem with nonmaintenance activities to minimise the combination of the total resource allocation cost, the weighted number of tardy jobs, and slack due-window assignment costs. A pseudo-polynomial time algorithm is designed to solve the bi-criteria scheduling problem.

Hadamard Semidifferential, Oriented Distance Function, and some Applications

<p style='text-indent:20px;'>The <i>Hadamard semidifferential calculus</i> preserves all the operations of the classical differential calculus including the chain rule for a large family of non-differentiable functions including the continuous convex functions. It naturally extends from the <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional Euclidean space <inline-formula><tex-math id="M2">\begin{document}$ \operatorname{\mathbb R}^n $\end{document}</tex-math></inline-formula> to subsets of topological vector spaces. This includes most function spaces used in <i>Optimization</i> and the <i>Calculus of Variations</i>, the metric groups used in <i>Shape and Topological Optimization</i>, and functions defined on submanifolds.</p><p style='text-indent:20px;'>Certain set-parametrized functions such as the <i>characteristic function</i> <inline-formula><tex-math id="M3">\begin{document}$ \chi_A $\end{document}</tex-math></inline-formula>of a set <inline-formula><tex-math id="M4">\begin{document}$ A $\end{document}</tex-math></inline-formula>, the <i>distance function</i> <inline-formula><tex-math id="M5">\begin{document}$ d_A $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M6">\begin{document}$ A $\end{document}</tex-math></inline-formula>, and the <i>oriented (signed) distance function</i> <inline-formula><tex-math id="M7">\begin{document}$ b_A = d_A-d_{ \operatorname{\mathbb R}^n\backslash A} $\end{document}</tex-math></inline-formula> can be used to identify a space of subsets of <inline-formula><tex-math id="M8">\begin{document}$ \operatorname{\mathbb R}^n $\end{document}</tex-math></inline-formula> with a metric space of set-parametrized functions. Many geometrical properties of domains (convexity, outward unit normal, curvatures, tangent space, smoothness of boundaries) can be expressed in terms of the analytical properties of <inline-formula><tex-math id="M9">\begin{document}$ b_A $\end{document}</tex-math></inline-formula> and a simple intrinsic differential calculus is available for functions defined on hypersurfaces without appealing to local bases or Christoffel symbols.</p><p style='text-indent:20px;'>The object of this paper is to extend the use of the Hadamard semidifferential and of the oriented distance function from finite to infinite dimensional spaces with some selected illustrative applications from shapes and geometries, plasma physics, and optimization.</p>

On the Weak Convergence of Monge-Ampère Measures for Discrete Convex Mesh Functions

To a mesh function we associate the natural analogue of the Monge-Ampere measure. The latter is shown to be equivalent to the Monge-Ampere measure of the convex envelope. We prove that the uniform convergence to a bounded convex function of mesh functions implies the uniform convergence on compact subsets of their convex envelopes and hence the weak convergence of the associated Monge-Ampere measures. We also give conditions for mesh functions to have a subsequence which converges uniformly to a convex function. Our result can be used to give alternate proofs of the convergence of some discretizations for the second boundary value problem for the Monge-Ampere equation and was used for a recently proposed discretization of the latter. For mesh functions which are uniformly bounded and satisfy a convexity condition at the discrete level, we show that there is a subsequence which converges uniformly on compact subsets to a convex function. The convex envelopes of the mesh functions of the subsequence also converge uniformly on compact subsets. If in addition they agree with a continuous convex function on the boundary, the limit function is shown to satisfy the boundary condition strongly.

Refinements of Hermite–Hadamard Inequalities for Continuous Convex Functions via (p,q)-Calculus

In this paper, we present some new refinements of Hermite–Hadamard inequalities for continuous convex functions by using (p,q)-calculus. Moreover, we study some new (p,q)-Hermite–Hadamard inequalities for multiple integrals. Many results given in this paper provide extensions of others given in previous research.

Projection/fixed point method for solving a semilinear obstacle problem

In this paper, we present a reformulation of a unilateral semilinear obstacle problem as a projection/fixed point problem based on appropriate variational inequality of the second kind and the subdifferential μ of a convex continuous function. The function μ leads to the characterization of the contact domain. Then we present the algorithms to solve the reformulated problem. We approximate the continuous problem by finite element method, then we present the analysis of the discrete problem and prove the convergence of the approximate solutions to the exact one.

From the Hahn–Banach extension theorem to the isotonicity of convex functions and the majorization theory

The property of isotonicity of a continuous convex function on the positive cone is characterized via subdifferentials. This is used to illustrate a new generalization of the Hardy–Littlewood–Polya inequality of majorization to the case of functions of a vector variable.

Relatively optimal production-distribution plans for stock management in supply chains over a finite time horizon

A model of the centralized dynamic production - distribution control is considered. In a supply chain a vendor supplies a product to a buyer, sending the products to the buyer’s warehouse in equal-size batches. Costs are attached to manufacturing the batch, the delivery of shipment and stockholding at both agents, and defined on production-distribution cycles (PDCs). There is only one production setup in a cycle with the restriction that in each warehouse the beginning and the ending inventory levels are the same. It is shown that the average cost of “sensible” PDCs is a continuous piece-wise convex function of their production batch sizes. For infinite time horizon the optimal production distribution plan is a constant sequence of PDCs with lowest average cost - economic production-distribution cycles (EPDCs).

On the global shape of convex functions on locally convex spaces

In the recent paper [1] D. Azagra studies the global shape of continuous convex functions defined on a Banach space X. More precisely, when X is separable, it is shown that for every continuous convex function f:X→R there exist a unique closed linear subspace Y of X, a convex function h:X/Y→R with the property that limt→∞⁡h(u+tv)=∞ for all u,v∈X/Y, v≠0, and x⁎∈X⁎ such that f=h∘π+x⁎, where π:X→X/Y is the natural projection. Our aim is to characterize those proper lower semicontinuous convex functions defined on a locally convex space which have the above representation. In particular, we show that the continuity of the function f and the completeness of X can be removed from the hypothesis of Azagra's theorem. For achieving our goal we study general sublinear functions as well as recession functions associated to convex ones.

Nonlinear Eigenvector Methods for Convex Minimization over the Numerical Range

We consider the optimization problem in which a continuous convex function is to be minimized over the joint numerical range of two Hermitian matrices. When those matrices are of large size, solvin...

On the refinement of quantum Hermite-Hadamard inequalities for continuous convex functions

On the refinement of quantum Hermite-Hadamard inequalities for continuous convex functions

Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations

This article concerns the expressive power of depth in neural nets with ReLU activations and a bounded width. We are particularly interested in the following questions: What is the minimal width w min ( d ) so that ReLU nets of width w min ( d ) (and arbitrary depth) can approximate any continuous function on the unit cube [ 0 , 1 ] d arbitrarily well? For ReLU nets near this minimal width, what can one say about the depth necessary to approximate a given function? We obtain an essentially complete answer to these questions for convex functions. Our approach is based on the observation that, due to the convexity of the ReLU activation, ReLU nets are particularly well suited to represent convex functions. In particular, we prove that ReLU nets with width d + 1 can approximate any continuous convex function of d variables arbitrarily well. These results then give quantitative depth estimates for the rate of approximation of any continuous scalar function on the d-dimensional cube [ 0 , 1 ] d by ReLU nets with width d + 3 .

The Hilbert Transform and Orthogonal Martingales in Banach Spaces

Abstract Let $X$ be a given Banach space, and let $M$ and $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. The paper contains the proof of the estimate $\mathbb E \Psi (N_t) \leq C_{\Phi ,\Psi ,X} \mathbb E \Phi (M_t)$, $t\geq 0$, where $\Phi , \Psi :X \to \mathbb R_+$ are convex continuous functions and the least admissible constant $C_{\Phi ,\Psi ,X}$ coincides with the $\Phi ,\Psi $-norm of the periodic Hilbert transform. As a corollary, it is shown that the $\Phi ,\Psi $-norms of the periodic Hilbert transform, the Hilbert transform on the real line, and the discrete Hilbert transform are the same if $\Phi $ is symmetric. We also prove that under certain natural assumptions on $\Phi $ and $\Psi $, the condition $C_{\Phi ,\Psi ,X}<\infty $ yields the UMD property of the space $X$. As an application, we provide comparison of $L^p$-norms of the periodic Hilbert transform to Wiener and Paley–Walsh decoupling constants. We also study the norms of the periodic, nonperiodic, and discrete Hilbert transforms and present the corresponding estimates in the context of differentially subordinate harmonic functions and more general singular integral operators.