Ting Kei Pong
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
Ting Kei Pong has a total of 67 co-authors. Here are the top co-authors by number of shared publications:
- Guoyin Li
Department of Applied Mathematics, University of New South Wales, Sydney, Australia
14 shared publications - Akiko Takeda
Graduate School of Information Science and Technology, University of Tokyo, Tokyo, Japan
9 shared publications - Bruno F. Lourenco
Department of Fundamental Statistical Mathematics, Institute of Statistical Mathematics, Tokyo, Japan
8 shared publications - Scott Boivin Lindstrom
Curtin University
7 shared publications - Xiaojun Chen
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China
7 shared publications - Tianxiang Liu
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
6 shared publications
Latest Publications
- Open Access
Complexity and convergence analysis of a single-loop SDCAM for Lipschitz composite optimization and beyond
We develop and analyze a single-loop algorithm for minimizing the sum of a Lipschitz differentiable function $f$, a prox-friendly proper closed function $g$ (with a closed domain on which $g$ is continuous) and the composition of another prox-friendly proper closed function $h$ (whose domain is closed on which $h$ is continuous) with a continuously differentiable mapping $c$ (that is Lipschitz continuous and Lipschitz differentiable on the convex closure of the domain of $g$). Such models arise naturally in many contemporary applications, where $f$ is the loss function for data misfit, and $g$ and $h$ are nonsmooth functions for inducing desirable structures in $x$ and $c(x)$. Existing single-loop algorithms mainly focus either on the case where $h$ is Lipschitz continuous or the case where $h$ is an indicator function of a closed convex set. In this paper, we develop a single-loop algorithm for more general possibly non-Lipschitz $h$. Our algorithm is a single-loop variant of the successive difference-of-convex approximation method (SDCAM) proposed in [22]. We show that when $h$ is Lipschitz, our algorithm exhibits an iteration complexity that matches the best known complexity result for obtaining an $(ε_1,ε_2,0)$-stationary point. Moreover, we show that, by assuming additionally that dom $g$ is compact, our algorithm exhibits an iteration complexity of $\tilde{O}(ε^{-4})$ for obtaining an $(ε,ε,ε)$-stationary point when $h$ is merely continuous and real-valued. Furthermore, we consider a scenario where $h$ does not have full domain and establish vanishing bounds on successive changes of iterates. Finally, in all three cases mentioned above, we show that one can construct a subsequence such that any accumulation point $x^*$ satisfies $c(x^*)\in$ dom $h$, and if a standard constraint qualification holds at $x^*$, then $x^*$ is a stationary point.Open AccessComplexity and convergence analysis of a single-loop SDCAM for Lipschitz composite optimization and beyond
We develop and analyze a single-loop algorithm for minimizing the sum of a Lipschitz differentiable function $f$, a prox-friendly proper closed function $g$ (with a closed domain on which $g$ is continuous) and the composition of another prox-friendly proper closed function $h$ (whose domain is closed on which $h$ is continuous) with a continuously differentiable mapping $c$ (that is Lipschitz continuous and Lipschitz differentiable on the convex closure of the domain of $g$). Such models arise naturally in many contemporary applications, where $f$ is the loss function for data misfit, and $g$ and $h$ are nonsmooth functions for inducing desirable structures in $x$ and $c(x)$. Existing single-loop algorithms mainly focus either on the case where $h$ is Lipschitz continuous or the case where $h$ is an indicator function of a closed convex set. In this paper, we develop a single-loop algorithm for more general possibly non-Lipschitz $h$. Our algorithm is a single-loop variant of the successive difference-of-convex approximation method (SDCAM) proposed in [22]. We show that when $h$ is Lipschitz, our algorithm exhibits an iteration complexity that matches the best known complexity result for obtaining an $(ε_1,ε_2,0)$-stationary point. Moreover, we show that, by assuming additionally that dom $g$ is compact, our algorithm exhibits an iteration complexity of $\tilde{O}(ε^{-4})$ for obtaining an $(ε,ε,ε)$-stationary point when $h$ is merely continuous and real-valued. Furthermore, we consider a scenario where $h$ does not have full domain and establish vanishing bounds on successive changes of iterates. Finally, in all three cases mentioned above, we show that one can construct a subsequence such that any accumulation point $x^*$ satisfies $c(x^*)\in$ dom $h$, and if a standard constraint qualification holds at $x^*$, then $x^*$ is a stationary point. - Open Access
Error bounds for perspective cones of a class of nonnegative Legendre functions
Error bounds play a central role in the study of conic optimization problems, including the analysis of convergence rates for numerous algorithms. Curiously, those error bounds are often Hölderian with exponent 1/2. In this paper, we try to explain the prevalence of the 1/2 exponent by investigating generic properties of error bounds for conic feasibility problems where the underlying cone is a perspective cone constructed from a nonnegative Legendre function on $\mathbb{R}$. Our analysis relies on the facial reduction technique and the computation of one-step facial residual functions (1-FRFs). Specifically, under appropriate assumptions on the Legendre function, we show that 1-FRFs can be taken to be Hölderian of exponent 1/2 almost everywhere with respect to the two-dimensional Hausdorff measure. This enables us to further establish that having a uniform Hölderian error bound with exponent 1/2 is a generic property for a class of feasibility problems involving these cones.Open AccessError bounds for perspective cones of a class of nonnegative Legendre functions
Error bounds play a central role in the study of conic optimization problems, including the analysis of convergence rates for numerous algorithms. Curiously, those error bounds are often Hölderian with exponent 1/2. In this paper, we try to explain the prevalence of the 1/2 exponent by investigating generic properties of error bounds for conic feasibility problems where the underlying cone is a perspective cone constructed from a nonnegative Legendre function on $\mathbb{R}$. Our analysis relies on the facial reduction technique and the computation of one-step facial residual functions (1-FRFs). Specifically, under appropriate assumptions on the Legendre function, we show that 1-FRFs can be taken to be Hölderian of exponent 1/2 almost everywhere with respect to the two-dimensional Hausdorff measure. This enables us to further establish that having a uniform Hölderian error bound with exponent 1/2 is a generic property for a class of feasibility problems involving these cones. - 1 May 2025
- Published by Institute for Operations Research and the Management Sciences (INFORMS) in Mathematics of Operations Research
- Vol. 50 (2) , 1204-1232
Optimal Error Bounds in the Absence of Constraint Qualifications with Applications to p-Cones and Beyond
We prove tight Hölderian error bounds for all p-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured. Moreover, they illuminate p-cones as a curious example of a class of objects that possess properties in three dimensions that they do not in four or more. Using our error bounds, we analyse least squares problems with p-norm regularization, where our results enable us to compute the corresponding Kurdyka–Łojasiewicz exponents for previously inaccessible values of p. Another application is a (relatively) simple proof that most p-cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions, and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight. Funding: The second author was partly supported by the Japan Society for the Promotion of Science Grant-in-Aid for Early-Career Scientists [Grants 19K20217, 23K16844] and Grant-in-Aid for Scientific Research [Grant (B)21H03398]. The third author was partly supported by the Hong Kong Research Grants Council [Grant PolyU153000/20p].
- 1 May 2025
- Published by Institute for Operations Research and the Management Sciences (INFORMS) in Mathematics of Operations Research
- Vol. 50 (2) , 1204-1232
Optimal Error Bounds in the Absence of Constraint Qualifications with Applications to p-Cones and Beyond
We prove tight Hölderian error bounds for all p-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured. Moreover, they illuminate p-cones as a curious example of a class of objects that possess properties in three dimensions that they do not in four or more. Using our error bounds, we analyse least squares problems with p-norm regularization, where our results enable us to compute the corresponding Kurdyka–Łojasiewicz exponents for previously inaccessible values of p. Another application is a (relatively) simple proof that most p-cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions, and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight. Funding: The second author was partly supported by the Japan Society for the Promotion of Science Grant-in-Aid for Early-Career Scientists [Grants 19K20217, 23K16844] and Grant-in-Aid for Scientific Research [Grant (B)21H03398]. The third author was partly supported by the Hong Kong Research Grants Council [Grant PolyU153000/20p].
- Open Access
- 9 April 2025
- Published by Springer Nature in Journal of Optimization Theory and Applications
- Vol. 205 (3) , 1-42
Tight Error Bounds for Log-Determinant Cones Without Constraint Qualifications
In this paper, without requiring any constraint qualifications, we establish tight error bounds for the log-determinant cone, which is the closure of the hypograph of the perspective function of the log-determinant function. This error bound is obtained using the recently developed framework based on one-step facial residual functions.
- 9 April 2025
- Published by Springer Nature in Journal of Optimization Theory and Applications
- Vol. 205 (3) , 1-42
Open AccessTight Error Bounds for Log-Determinant Cones Without Constraint Qualifications
In this paper, without requiring any constraint qualifications, we establish tight error bounds for the log-determinant cone, which is the closure of the hypograph of the perspective function of the log-determinant function. This error bound is obtained using the recently developed framework based on one-step facial residual functions.
- Open Access
- 7 April 2025
- Published by Springer Nature in Computational Optimization and Applications
- Vol. 91 (3) , 1185-1225
An extended sequential quadratic method with extrapolation
We revisit and adapt the extended sequential quadratic method (ESQM) in Auslender (J Optim Theory Appl 156:183–212, 2013) for solving a class of difference-of-convex optimization problems whose constraints are defined as the intersection of level sets of Lipschitz differentiable functions and a simple compact convex set. Particularly, for this class of problems, we develop a variant of ESQM, called ESQM with extrapolation ( $$\hbox {ESQM}_{\textrm{e}}$$ ), which incorporates Nesterov’s extrapolation techniques for empirical acceleration. Under standard constraint qualifications, we show that the sequence generated by $$\hbox {ESQM}_{\textrm{e}}$$ clusters at a critical point if the extrapolation parameters are uniformly bounded above by a certain threshold. Convergence of the whole sequence and the convergence rate are established by assuming Kurdyka-Łojasiewicz (KL) property of a suitable potential function and imposing additional differentiability assumptions on the objective and constraint functions. In addition, when the objective and constraint functions are all convex, we show that linear convergence can be established if a certain exact penalty function is known to be a KL function with exponent $$\frac{1}{2}$$ ; we also discuss how the KL exponent of such an exact penalty function can be deduced from that of the original extended objective (i.e., sum of the objective and the indicator function of the constraint set). Finally, we perform numerical experiments to demonstrate the empirical acceleration of $$\hbox {ESQM}_{\textrm{e}}$$ over a basic version of ESQM, and illustrate its effectiveness by comparing with the natural competing algorithm $$\hbox {SCP}_{\textrm{ls}}$$ from Yu et al. (SIAM J Optim 31:2024–2054, 2021).- 7 April 2025
- Published by Springer Nature in Computational Optimization and Applications
- Vol. 91 (3) , 1185-1225
Open AccessAn extended sequential quadratic method with extrapolation
We revisit and adapt the extended sequential quadratic method (ESQM) in Auslender (J Optim Theory Appl 156:183–212, 2013) for solving a class of difference-of-convex optimization problems whose constraints are defined as the intersection of level sets of Lipschitz differentiable functions and a simple compact convex set. Particularly, for this class of problems, we develop a variant of ESQM, called ESQM with extrapolation ( $$\hbox {ESQM}_{\textrm{e}}$$ ), which incorporates Nesterov’s extrapolation techniques for empirical acceleration. Under standard constraint qualifications, we show that the sequence generated by $$\hbox {ESQM}_{\textrm{e}}$$ clusters at a critical point if the extrapolation parameters are uniformly bounded above by a certain threshold. Convergence of the whole sequence and the convergence rate are established by assuming Kurdyka-Łojasiewicz (KL) property of a suitable potential function and imposing additional differentiability assumptions on the objective and constraint functions. In addition, when the objective and constraint functions are all convex, we show that linear convergence can be established if a certain exact penalty function is known to be a KL function with exponent $$\frac{1}{2}$$ ; we also discuss how the KL exponent of such an exact penalty function can be deduced from that of the original extended objective (i.e., sum of the objective and the indicator function of the constraint set). Finally, we perform numerical experiments to demonstrate the empirical acceleration of $$\hbox {ESQM}_{\textrm{e}}$$ over a basic version of ESQM, and illustrate its effectiveness by comparing with the natural competing algorithm $$\hbox {SCP}_{\textrm{ls}}$$ from Yu et al. (SIAM J Optim 31:2024–2054, 2021). - Open Access
Convergence analysis for a variant of manifold proximal point algorithm based on Kurdyka-Łojasiewicz property
We incorporate an iteratively reweighted strategy in the manifold proximal point algorithm (ManPPA) in [12] to solve an enhanced sparsity inducing model for identifying sparse yet nonzero vectors in a given subspace. We establish the global convergence of the whole sequence generated by our algorithm by assuming the Kurdyka-Lojasiewicz (KL) properties of suitable potential functions. We also study how the KL exponents of the different potential functions are related. More importantly, when our enhanced model and algorithm reduce, respectively, to the model and ManPPA with constant stepsize considered in [12], we show that the sequence generated converges linearly as long as the optimal value of the model is positive, and converges finitely when the limit of the sequence lies in a set of weak sharp minima. Our results improve [13, Theorem 2.4], which asserts local quadratic convergence in the presence of weak sharp minima when the constant stepsize is small.
Open AccessConvergence analysis for a variant of manifold proximal point algorithm based on Kurdyka-Łojasiewicz property
We incorporate an iteratively reweighted strategy in the manifold proximal point algorithm (ManPPA) in [12] to solve an enhanced sparsity inducing model for identifying sparse yet nonzero vectors in a given subspace. We establish the global convergence of the whole sequence generated by our algorithm by assuming the Kurdyka-Lojasiewicz (KL) properties of suitable potential functions. We also study how the KL exponents of the different potential functions are related. More importantly, when our enhanced model and algorithm reduce, respectively, to the model and ManPPA with constant stepsize considered in [12], we show that the sequence generated converges linearly as long as the optimal value of the model is positive, and converges finitely when the limit of the sequence lies in a set of weak sharp minima. Our results improve [13, Theorem 2.4], which asserts local quadratic convergence in the presence of weak sharp minima when the constant stepsize is small.
- 1 January 2025
- Published by Chongqing Normal University in Pacific Journal of Optimization
- Vol. 21 (4) , 761-785
Convergence analysis for a variant of manifold proximal point algorithm based on Kurdyka-Łojasiewicz property
(Communicated by Liqun Qi) (Dedicated to Professor Terry Rockafellar on the occasion of his 90th birthday) We incorporate an iteratively reweighted strategy in the manifold proximal point algorithm (ManPPA) in [1] to solve an enhanced sparsity inducing model for identifying sparse yet nonzero vectors in a given subspace. We establish the global convergence of the whole sequence generated by our algorithm by assuming the Kurdyka-Łojasiewicz (KŁ) properties of suitable potential functions. We also study how the KŁ exponents of the different potential functions are related. More importantly, when our enhanced model and algorithm reduce, respectively, to the model and ManPPA with constant stepsize considered in [1], we show that the sequence generated converges linearly as long as the optimal value of the model is positive, and converges finitely when the limit of the sequence lies in a set of weak sharp minima. Our results improve [2,Theorem 2.4], which asserts local quadratic convergence in the presence of weak sharp minima when the constant stepsize is small.
- 1 January 2025
- Published by Chongqing Normal University in Pacific Journal of Optimization
- Vol. 21 (4) , 761-785
Convergence analysis for a variant of manifold proximal point algorithm based on Kurdyka-Łojasiewicz property
(Communicated by Liqun Qi) (Dedicated to Professor Terry Rockafellar on the occasion of his 90th birthday) We incorporate an iteratively reweighted strategy in the manifold proximal point algorithm (ManPPA) in [1] to solve an enhanced sparsity inducing model for identifying sparse yet nonzero vectors in a given subspace. We establish the global convergence of the whole sequence generated by our algorithm by assuming the Kurdyka-Łojasiewicz (KŁ) properties of suitable potential functions. We also study how the KŁ exponents of the different potential functions are related. More importantly, when our enhanced model and algorithm reduce, respectively, to the model and ManPPA with constant stepsize considered in [1], we show that the sequence generated converges linearly as long as the optimal value of the model is positive, and converges finitely when the limit of the sequence lies in a set of weak sharp minima. Our results improve [2,Theorem 2.4], which asserts local quadratic convergence in the presence of weak sharp minima when the constant stepsize is small.
- Open Access
Break recovery in graphical networks with D-trace loss
We consider the problem of estimating a time-varying sparse precision matrix, which is assumed to evolve in a piece-wise constant manner. Building upon the Group Fused LASSO and LASSO penalty functions, we estimate both the network structure and the change-points. We propose an alternative estimator to the commonly employed Gaussian likelihood loss, namely the D-trace loss. We provide the conditions for the consistency of the estimated change-points and of the sparse estimators in each block. We show that the solutions to the corresponding estimation problem exist when some conditions relating to the tuning parameters of the penalty functions are satisfied. Unfortunately, these conditions are not verifiable in general, posing challenges for tuning the parameters in practice. To address this issue, we introduce a modified regularizer and develop a revised problem that always admits solutions: these solutions can be used for detecting possible unsolvability of the original problem or obtaining a solution of the original problem otherwise. An alternating direction method of multipliers (ADMM) is then proposed to solve the revised problem. The relevance of the method is illustrated through simulations and real data experiments.
Open AccessBreak recovery in graphical networks with D-trace loss
We consider the problem of estimating a time-varying sparse precision matrix, which is assumed to evolve in a piece-wise constant manner. Building upon the Group Fused LASSO and LASSO penalty functions, we estimate both the network structure and the change-points. We propose an alternative estimator to the commonly employed Gaussian likelihood loss, namely the D-trace loss. We provide the conditions for the consistency of the estimated change-points and of the sparse estimators in each block. We show that the solutions to the corresponding estimation problem exist when some conditions relating to the tuning parameters of the penalty functions are satisfied. Unfortunately, these conditions are not verifiable in general, posing challenges for tuning the parameters in practice. To address this issue, we introduce a modified regularizer and develop a revised problem that always admits solutions: these solutions can be used for detecting possible unsolvability of the original problem or obtaining a solution of the original problem otherwise. An alternating direction method of multipliers (ADMM) is then proposed to solve the revised problem. The relevance of the method is illustrated through simulations and real data experiments.
- Open Access
Tight error bounds for log-determinant cones without constraint qualifications
In this paper, without requiring any constraint qualifications, we establish tight error bounds for the log-determinant cone, which is the closure of the hypograph of the perspective function of the log-determinant function. This error bound is obtained using the recently developed framework based on one-step facial residual functions.
Open AccessTight error bounds for log-determinant cones without constraint qualifications
In this paper, without requiring any constraint qualifications, we establish tight error bounds for the log-determinant cone, which is the closure of the hypograph of the perspective function of the log-determinant function. This error bound is obtained using the recently developed framework based on one-step facial residual functions.
- Open Access
Optimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond
We prove tight H\"olderian error bounds for all $p$-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured; moreover, they illuminate $p$-cones as a curious example of a class of objects that possess properties in 3 dimensions that they do not in 4 or more. Using our error bounds, we analyse least squares problems with $p$-norm regularization, where our results enable us to compute the corresponding KL exponents for previously inaccessible values of $p$. Another application is a (relatively) simple proof that most $p$-cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions, and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight.Open AccessOptimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond
We prove tight H\"olderian error bounds for all $p$-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured; moreover, they illuminate $p$-cones as a curious example of a class of objects that possess properties in 3 dimensions that they do not in 4 or more. Using our error bounds, we analyse least squares problems with $p$-norm regularization, where our results enable us to compute the corresponding KL exponents for previously inaccessible values of $p$. Another application is a (relatively) simple proof that most $p$-cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions, and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight.