arXiv subjects

Chayne Planiden

Publications and source records attributed to Chayne Planiden.

18 recordsLinked to original sources

arXiv2026

The cosine measure of a function at a point

The cosine measure of a set of vectors in $\mathbb{R}^n$ measures how well the set covers all directions in $\mathbb{R}^n$. It identifies the direction furthest, in angle, from the set. It is used in the convergence theory of various optimization algorithms, but also highlights interesting geometric properties of sets. For example, the cosine measure of a set $S$ is greater than zero if and only if given any $\mathcal{C}^1$ function $f$ at a point $\mathbf{x}$ where the gradient is nonzero, $S$ must contain a descent direction of $f$ at $\mathbf{x}$. In this paper, we examine the question of what can be said when the function $f$ is non-differentiable or if it has a gradient equal to the zero vector. To examine these cases, we introduce the novel concept of the {\em cosine measure of a function} at a point. This value provides an infimum on the value of the cosine measure that a set of vectors requires to guarantee it contains a descent direction of the function at the point of interest. We present mathematical theory around this concept, including examples showing that the cosine measure of a smooth function can have any value in $[-1,1]$. We further present algorithms to compute the cosine measure of a function, and examples demonstrating the algorithm on smooth and nonsmooth functions. These results also shed light on the the cosine measure of infinite sets and nonconvex cones.

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arXiv2023

Using generalized simplex methods to approximate derivatives

This paper presents two methods for approximating a proper subset of the entries of a Hessian using only function evaluations. These approximations are obtained using the techniques called \emph{generalized simplex Hessian} and \emph{generalized centered simplex Hessian}. We show how to choose the matrices of directions involved in the computation of these two techniques depending on the entries of the Hessian of interest. We discuss the number of function evaluations required in each case and develop a general formula to approximate all order-$P$ partial derivatives. Since only function evaluations are required to compute the methods discussed in this paper, they are suitable for use in derivative-free optimization methods.

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arXiv2021

Nicely structured positive bases with maximal cosine measure

The properties of positive bases make them a useful tool in derivative-free optimization (DFO) and an interesting concept in mathematics. The notion of the \emph{cosine measure} helps to quantify the quality of a positive basis. It provides information on how well the vectors in the positive basis uniformly cover the space considered. The number of vectors in a positive basis is known to be between $n+1$ and $2n$ inclusively. When the number of vectors is strictly between $n+1$ and $2n$, we say that it is an intermediate positive basis. In this paper, the structure of intermediate positive bases with maximal cosine measure is investigated. The structure of an intermediate positive basis with maximal cosine measure over a certain subset of positive bases is provided. This type of positive bases has a simple structure that makes them easy to generate with a computer software.

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arXiv2021

Limiting behaviour of the generalized simplex gradient as the number of points tends to infinity on a fixed shape in R^n

This work investigates the asymptotic behaviour of the gradient approximation method called the generalized simplex gradient (GSG). This method has an error bound that at first glance seems to tend to infinity as the number of sample points increases, but with some careful construction, we show that this is not the case. For functions in finite dimensions, we present two new error bounds ad infinitum depending on the position of the reference point. The error bounds are not a function of the number of sample points and thus remain finite.

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arXiv2020

Hessian approximations

This work introduces the nested-set Hessian approximation, a second-order approximation method that can be used in any derivative-free optimization routine that requires such information. It is built on the foundation of the generalized simplex gradient and proved to have an error bound that is on the order of the maximal radius of the two sets used in its construction. We show that when the points used in the computation of the nested-set Hessian have a favourable structure, (n+1)(n+2)/2 function evaluations are sufficient to approximate the Hessian. However, the nested-set Hessian also allows for evaluation sets with more points without negating the error analysis. Two calculus-based approximation techniques of the Hessian are developed and some advantages of the same are demonstrated.

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arXiv2020

Error bounds for overdetermined and underdetermined generalized centred simplex gradients

Using the Moore--Penrose pseudoinverse, this work generalizes the gradient approximation technique called centred simplex gradient to allow sample sets containing any number of points. This approximation technique is called the \emph{generalized centred simplex gradient}. We develop error bounds and, under a full-rank condition, show that the error bounds have order $O(Δ^2)$, where $Δ$ is the radius of the sample set of points used. We establish calculus rules for generalized centred simplex gradients, introduce a calculus-based generalized centred simplex gradient and confirm that error bounds for this new approach are also order $O(Δ^2)$. We provide several examples to illustrate the results and some benefits of these new methods.

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arXiv2019

Parametrically prox-regular functions

Prox-regularity is a generalization of convexity that includes all C2, lower-C2, strongly amenable and primal-lower-nice functions. The study of prox-regular functions provides insight on a broad spectrum of important functions. Parametrically prox-regular (para-prox-regular) functions are a further extension of this family, produced by adding a parameter. Such functions have been shown to play a key role in understanding stability of minimizers in optimization problems. This document discusses para-prox-regular functions in Rn: we begin with some basic examples of para-prox-regular functions and move on to the more complex examples of the convex and nonconvex proximal average. We develop an alternate representation of a para-prox-regular function, related to the monotonicity of an f-attentive epsilon-localization as has been done for prox-regular functions. This extends a result of Levy, who used an alternate approach to show one implication of the relationship (we provide a characterization). We analyze two common forms of parametrized functions that appear in optimization: finite parametrized sum of functions and finite parametrized max of functions. The example of strongly amenable functions by Poliquin and Rockafellar is given and a relaxation of its necessary conditions is presented.

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arXiv2019

Conditions for the existence, identification and calculus rules of the threshold of prox-boundedness

This work advances knowledge of the threshold of prox-boundedness of a function; an important concern in the use of proximal point optimization algorithms and in determining the existence of the Moreau envelope of the function. In finite dimensions, we study general prox-bounded functions and then focus on some useful families such as piecewise functions and Lipschitz continuous functions. The thresholds are explicitly determined when possible and bounds are established otherwise. Some calculus rules are constructed; we consider functions with known thresholds and find the thresholds of their sum and composition.

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arXiv2019

The chain rule for VU-decompositions of nonsmooth functions

In Variational Analysis, VU-theory provides a set of tools that is helpful for understanding and exploiting the structure of nonsmooth functions. The theory takes advantage of the fact that at any point, the space can be separated into two orthogonal subspaces: one that describes the direction of nonsmoothness of the function, and the other on which the function behaves smoothly and has a gradient. For a composite function, this work establishes a chain rule that facilitates the computation of such gradients and characterizes the smooth subspace under reasonable conditions. From the chain rule presented, formulas for the separation, smooth perturbation and sum of functions are provided. Several nonsmooth examples are explored, including norm functions, max-of-quadratic functions and LASSO-type regularizations.

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arXiv2019

Proximal mappings and Moreau envelopes of single-variable convex piecewise cubic functions and multivariable gauge functions

This work presents a collection of useful properties of the Moreau envelope for finite-dimensional, proper, lower semicontinuous, convex functions. In particular, gauge functions and piecewise cubic functions are investigated and their Moreau envelopes categorized. Characterizations of convex Moreau envelopes are established; topics include strict convexity, strong convexity and Lipschitz continuity.

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arXiv2019

Epiconvergence, the Moreau envelope and generalized linear-quadratic functions

This work introduces the class of generalized linear-quadratic functions, constructed using maximally monotone symmetric linear relations. Calculus rules and properties of the Moreau envelope for this class of functions are developed. In finite dimensions, on a metric space defined by Moreau envelopes, we consider the epigraphical limit of a sequence of quadratic functions and categorize the results. We explore the question of when a quadratic function is a Moreau envelope of a generalized linear-quadratic function; characterizations involving nonexpansiveness and Lipschitz continuity are established. This work generalizes some results by Hiriart-Urruty and by Rockafellar and Wets.

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arXiv2019

The NC-proximal average for multiple functions

The NC-proximal average is a parametrized function used to continuously transform one proper, lsc, prox-bounded function into another. Until now, it has been defined for two functions. The purpose of this article is to redefine it so that any finite number of functions may be used. The layout generally follows that of [11], extending those results to the more general case and in some instances giving alternate proofs by using techniques developed after the publication of that paper. We conclude with an example examining the discontinuity of the minimizers of the NC-proximal average.

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arXiv2019

Some geometric properties of the solutions of complex multi-affine polynomials of degree three

In this paper, we consider complex polynomials of degree three with distinct zeros and their polarization ((z1,z2,z3) with three complex variables. We show, through elementary means, that the variety P(z1,z2,z3)=0 is birationally equivalent to the variety z1z2z3 +1 = 0. Moreover, the rational map certifying the equivalence is a simple Möbius transformation. The second goal of this note is to present a geometrical curiosity relating the zeros of P(z,z,zk) for k = 1,2,3, where (z1,z2,z3) is arbitrary point on the variety P(z1,z2 z3) = 0.

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arXiv2019

A proximal average for prox-bounded functions

In this work, we construct a proximal average for two prox-bounded functions, which recovers the classical proximal average for two convex functions. The new proximal average transforms continuously in epi-topology from one proximal hull to the other. When one of the functions is differentiable, the new proximal average is differentiable. We give characterizations for Lipschitz and single-valued proximal mappings and we show that the convex combination of convexified proximal mappings is always a proximal mapping. Subdifferentiability and behaviors of infimal values and minimizers are also studied.

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arXiv2019

A derivative-free $\mathcal{VU}$-algorithm for convex finite-max problems

The $\mathcal{VU}$-algorithm is a superlinearly convergent method for minimizing nonsmooth, convex functions. At each iteration, the algorithm works with a certain $\mathcal{V}$-space and its orthogonal $\U$-space, such that the nonsmoothness of the objective function is concentrated on its projection onto the $\mathcal{V}$-space, and on the $\mathcal{U}$-space the projection is smooth. This structure allows for an alternation between a Newton-like step where the function is smooth, and a proximal-point step that is used to find iterates with promising $\mathcal{VU}$-decompositions. We establish a derivative-free variant of the $\mathcal{VU}$-algorithm for convex finite-max objective functions. We show global convergence and provide numerical results from a proof-of-concept implementation, which demonstrates the feasibility and practical value of the approach. We also carry out some tests using nonconvex functions and discuss the results.

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arXiv2016

Computing proximal points of convex functions with inexact subgradients

Locating proximal points is a component of numerous minimization algorithms. This work focuses on developing a method to find the proximal point of a convex function at a point, given an inexact oracle. Our method assumes that exact function values are at hand, but exact subgradients are either not available or not useful. We use approximate subgradients to build a model of the objective function, and prove that the method converges to the true prox-point within acceptable tolerance. The subgradient $g_k$ used at each step $k$ is such that the distance from $g_k$ to the true subdifferential of the objective function at the current iteration point is bounded by some fixed $\varepsilon>0.$ The algorithm includes a novel tilt-correct step applied to the approximate subgradient.

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arXiv2016

Thresholds of Prox-Boundedness of PLQ functions

Introduced in the 1960s, the Moreau envelope has grown to become a key tool in non\-smooth analysis and optimization. Essentially an infimal convolution with a parametrized norm squared, the Moreau envelope is used in many applications and optimization algorithms. An important aspect in applying the Moreau envelope to nonconvex functions is determining if the function is prox-bounded, that is, if there exists a point $x$ and a parameter $r$ such that the Moreau envelope is finite. The infimum of all such $r$ is called the threshold of prox-boundedness (prox-threshold) of the function $f.$ In this paper, we seek to understand the prox-thresholds of piecewise linear-quadratic (PLQ) functions. (A PLQ function is a function whose domain is a union of finitely many polyhedral sets, and that is linear or quadratic on each piece.) The main result provides a computational technique for determining the prox-threshold for a PLQ function, and further analyzes the behavior of the Moreau envelope of the function using the prox-threshold. We provide several examples to illustrate the techniques and challenges.

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arXiv2015

Strongly convex functions, Moreau envelopes and the generic nature of convex functions with strong minimizers

In this work, using Moreau envelopes, we define a complete metric for the set of proper lower semicontinuous convex functions. Under this metric, the convergence of each sequence of convex functions is epi-convergence. We show that the set of strongly convex functions is dense but it is only of the first category. On the other hand, it is shown that the set of convex functions with strong minima is of the second category.

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