# Chayne PlanidenUniversity of Wollongong | UOW · School of Mathematics and Applied Statistics (SMAS)

Chayne Planiden

Ph. D. Mathematics

## About

31

Publications

3,885

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103

Citations

Citations since 2016

Introduction

Chayne Planiden currently works at the Department of Mathematics and Applied Statistics, University of Wollongong. Chayne does research in Applied Mathematics and Analysis. His current project is 'Derivative-free Methods.'

Additional affiliations

February 2018 - present

January 2015 - January 2018

July 2014 - present

Education

September 2013 - June 2017

September 2011 - June 2013

September 2005 - June 2009

## Publications

Publications (31)

This work presents a novel matrix-based method for constructing an approximation Hessian using only function evaluations. The method requires less computational power than interpolation-based methods and is easy to implement in matrix-based programming languages such as MATLAB. As only function evaluations are required, the method is suitable for u...

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...

Global electricity markets are undergoing a rapid transformation in their energy mix to meet commitments towards sustainable electric grids. This change in energy mix engenders significant challenges, specifically concerning the management of non-dispatchable energy resources. System and market operators are required to meet power system security a...

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 dim...

This note 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 classes such as piecewise func...

Using the Moore–Penrose pseudoinverse this work generalizes the gradient approximation technique called the centred simplex gradient to allow sample sets containing any number of points. This approximation technique is called the generalized centred simplex gradient. We develop error bounds and, under a full-rank condition, show that the error boun...

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 use...

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...

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 characterizatio...

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 pa...

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 sequen...

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...

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 fun...

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 M\"obius tra...

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 c...

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...

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 whic...

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 $\math...

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...

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 whic...

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 subgra...

This work explores 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...

The VU-algorithm is a superlinearly convergent method for minimizing nonsmooth convex functions. At each iteration, the algorithm separates R n into the V-space and the orthogonal U-space, such that the nonsmoothness of the objective function is concentrated on its projection onto the V-space, and on the U-space the projection is smooth. This struc...

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...

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 tha...

In this paper we consider complex polynomials p(z) of degree three with distinct zeros and their polarization P(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 t...

Introduced in the 1960s, the Moreau envelope has grown to become a key tool in nonsmooth 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 determini...

Finding the minimum and the minimizers of convex functions has been of
primary concern in convex analysis since its conception. It is well-known that
if a convex function has a minimum, then that minimum is global. The
minimizers, however, may not be unique. There are certain subclasses, such as
strictly convex functions, that do have unique minimi...

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-proxregular)
functions are a further extension of this family, produced by adding a pa...

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 [Hare 2009], extending
those re...