arXiv subjects

Xin-Wei Liu

Publications and source records attributed to Xin-Wei Liu.

17 recordsLinked to original sources

arXiv2026

Polynomial iteration complexity of a path-following smoothing Newton method for symmetric cone programming

It has long remained open whether smoothing Newton methods (SNMs) for symmetric cone programming (SCP) admit polynomial iteration complexity. A key difficulty lies in the lack of an analogue of the self-concordant convex framework underlying interior-point methods (IPMs). In this paper, inspired by Nemirovski's self-concordant convex-concave theory, we address this open problem by introducing a reduced barrier augmented Lagrangian (BAL) function. We prove that the reduced BAL function is self-concordant convex-concave and establish that the parameterized smooth system arising in SNMs coincides with the first-order optimality conditions of an associated minimax problem. Motivated by this equivalence, we propose a path-following smoothing Newton method (PFSNM). The reduced BAL function induces a central path and an associated neighborhood, which provide estimates for the Newton decrement needed for the path-following analysis. As a result, the method achieves an iteration complexity of $\mathcal{O}(\sqrtν\ln(1/\varepsilon))$, matching the best-known short-step complexity for IPMs. Numerical results on standard benchmarks show that PFSNM is competitive with several well-known interior-point solvers, and the observed performance is consistent with the theoretical development.

math.OC↗

arXiv2026

A Newton Augmented Lagrangian Method for Symmetric Cone Programming with Complexity Analysis

Symmetric cone programming covers a broad class of convex optimization problems, including linear programming, second-order cone programming, and semidefinite programming. Although the augmented Lagrangian method (ALM) is well-suited for large-scale problems, its subproblems are often not twice continuously differentiable, preventing the direct use of classical Newton methods. To address this issue, we observe that barrier functions used in interior-point methods (IPMs) naturally serve as effective smoothing terms to alleviate such nonsmoothness. By combining the strengths of ALM and IPMs, we construct a novel augmented Lagrangian function and subsequently develop a Newton augmented Lagrangian (NAL) method. By leveraging the self-concordance property of the barrier function, the proposed method is shown to achieve an $\mathcal{O}(1/ε)$ complexity bound. In addition, a spectral analysis reveals that the condition numbers of the Schur complement matrices arising in the NAL method are of order $\mathcal{O}(1/μ)$, which is better than the $\mathcal{O}(1/{μ^2})$ order of classical IPMs. This improvement is further illustrated by a heatmap of condition numbers. Numerical experiments conducted on standard benchmarks indicate that the NAL method exhibits significant performance improvements compared to several existing methods.

math.OC↗

arXiv2025

A Surrogate Value Function Formulation for Bilevel Optimization

The value function formulation captures the hierarchical nature of bilevel optimization through the optimal value function of the lower level problem, yet its implicit and nonsmooth characteristics pose significant analytical and computational difficulties. We introduce a surrogate value function formulation that replaces the intractable value function with an explicit surrogate derived from lower level stationarity conditions. This surrogate formulation preserves the essential idea of the classical value function model but fundamentally departs from Karush Kuhn Tucker (KKT) formulations, which embed lower level stationary points into the upper level feasible region and obscure the hierarchical dependence. Instead, it enforces the hierarchy through a dominance constraint that remains valid even when lower level constraint qualifications fail at the solution. We establish equivalence with the original bilevel problem, reveal the failure of standard constraint qualifications, and show that its strong stationarity implies that of KKT models. To handle the complementarity constraints in the surrogate formulation, we apply a smoothing barrier augmented Lagrangian method and prove its convergence to solutions and Clarke stationary points. Extensive experiments demonstrate the robustness and high numerical precision of this formulation, especially in nonconvex settings, including the classical Mirrlees problem where KKT models fail.

math.OC↗

arXiv2024

Enhanced Barrier-Smoothing Technique for Bilevel Optimization with Nonsmooth Mappings

Bilevel optimization problems, encountered in fields such as economics, engineering, and machine learning, pose significant computational challenges due to their hierarchical structure and constraints at both upper and lower levels. Traditional gradient-based methods are effective for unconstrained bilevel programs with unique lower level solutions, but struggle with constrained bilevel problems due to the nonsmoothness of lower level solution mappings. To overcome these challenges, this paper introduces the Enhanced Barrier-Smoothing Algorithm (EBSA), a novel approach that integrates gradient-based techniques with an augmented Lagrangian framework. EBSA utilizes innovative smoothing functions to approximate the primal-dual solution mapping of the lower level problem, and then transforms the bilevel problem into a sequence of smooth single-level problems. This approach not only addresses the nonsmoothness but also enhances convergence properties. Theoretical analysis demonstrates its superiority in achieving Clarke and, under certain conditions, Bouligand stationary points for bilevel problems. Both theoretical analysis and preliminary numerical experiments confirm the robustness and efficiency of EBSA.

math.OC↗

arXiv2024

A mechanism of three-dimensional quadratic termination for the gradient method with applications

Recent studies show that the two-dimensional quadratic termination property has great potential in improving performance of the gradient method. However, it is not clear whether higher-dimensional quadratic termination leads further benefits. In this paper, we provide an affirmative answer by introducing a mechanism of three-dimensional quadratic termination for the gradient method. A novel stepsize is derived from the mechanism such that a family of delayed gradient methods equipping with the novel stepsize have the three-dimensional quadratic termination property. When applied to the Barzilai--Borwein (BB) method, the novel stepsize does not require the use of any exact line search or the Hessian, and can be computed by stepsizes and gradient norms in previous iterations. Using long BB steps and some short steps associated with the novel stepsize in an adaptive manner, we develop an efficient gradient method for quadratic optimization and further extend it to general unconstrained optimization. Numerical experiments show that the three-dimensional quadratic termination property can significantly improve performance of the BB method, and the proposed method outperforms gradient methods that use stepsizes with the two-dimensional quadratic termination property.

math.OC↗

arXiv2023

Exact penalty method for D-stationary point of nonlinear optimization

We consider the nonlinear optimization problem with least $\ell_1$-norm measure of constraint violations and introduce the concepts of the D-stationary point, the DL-stationary point and the DZ-stationary point with the help of exact penalty function. If the stationary point is feasible, they correspond to the Fritz-John stationary point, the KKT stationary point and the singular stationary point, respectively. In order to show the usefulness of the new stationary points, we propose a new exact penalty sequential quadratic programming (SQP) method with inner and outer iterations and analyze its global and local convergence. The proposed method admits convergence to a D-stationary point and rapid infeasibility detection without driving the penalty parameter to zero, which demonstrates the commentary given in [SIAM J. Optim., 20 (2010), 2281--2299] and can be thought to be a supplement of the theory of nonlinear optimization on rapid detection of infeasibility. Some illustrative examples and preliminary numerical results demonstrate that the proposed method is robust and efficient in solving infeasible nonlinear problems and a degenerate problem without LICQ in the literature.

math.OC↗

arXiv2022

A primal-dual interior-point relaxation method with global and rapidly local convergence for nonlinear programs

Based on solving an equivalent parametric equality constrained mini-max problem of the classic logarithmic-barrier subproblem, we present a novel primal-dual interior-point relaxation method for nonlinear programs with general equality and nonnegative constraints. In each iteration, our method approximately solves the KKT system of a parametric equality constrained mini-max subproblem, which avoids the requirement that any primal or dual iterate is an interior-point. The method has some similarities to the warmstarting interior-point methods in relaxing the interior-point requirement and is easily extended for solving problems with general inequality constraints. In particular, it has the potential to circumvent the jamming difficulty that appears with many interior-point methods for nonlinear programs and improve the ill conditioning of existing primal-dual interior-point methods as the barrier parameter is small. A new smoothing approach is introduced to develop our relaxation method and promote convergence of the method. Under suitable conditions, it is proved that our method can be globally convergent and locally quadratically convergent to the KKT point of the original problem. The preliminary numerical results on a well-posed problem for which many interior-point methods fail to find the minimizer and a set of test problems from the CUTEr collection show that our method is efficient.

math.OC↗

arXiv2022

A primal-dual majorization-minimization method for large-scale linear programs

We present a primal-dual majorization-minimization method for solving large-scale linear programs. A smooth barrier augmented Lagrangian (SBAL) function with strict convexity for the dual linear program is derived. The majorization-minimization approach is naturally introduced to develop the smoothness and convexity of the SBAL function. Our method only depends on a factorization of the constant matrix independent of iterations and does not need any computation on step sizes, thus can be expected to be particularly appropriate for large-scale linear programs. The method shares some similar properties to the first-order methods for linear programs, but its convergence analysis is established on the differentiability and convexity of our SBAL function. The global convergence is analyzed without prior requiring either the primal or dual linear program to be feasible. Under the regular conditions, our method is proved to be globally linearly convergent, and a new iteration complexity result is given.

math.OC↗

arXiv2021

A novel augmented Lagrangian method of multipliers for optimization with general inequality constraints

We introduce a twice differentiable augmented Lagrangian for nonlinear optimization with general inequality constraints and show that a strict local minimizer of the original problem is an approximate strict local solution of the augmented Lagrangian. A novel augmented Lagrangian method of multipliers (ALM) is then presented. Our method is originated from a generalization of the Hetenes-Powell augmented Lagrangian, and is a combination of the augmented Lagrangian and the interior-point technique. It shares a similar algorithmic framework with existing ALMs for optimization with inequality constraints, but it can use the second derivatives and does not depend on projections on the set of inequality constraints. In each iteration, our method solves a twice continuously differentiable unconstrained optimization subproblem on primal variables. The dual iterates, penalty and smoothing parameters are updated adaptively. The global and local convergence are analyzed. Without assuming any constraint qualification, it is proved that the proposed method has strong global convergence. The method may converge to either a Kurash-Kuhn-Tucker (KKT) point or a singular stationary point when the converging point is a minimizer. It may also converge to an infeasible stationary point of nonlinear program when the problem is infeasible. Furthermore, our method is capable of rapidly detecting the possible infeasibility of the solved problem. Under suitable conditions, it is locally linearly convergent to the KKT point, which is consistent with ALMs for optimization with equality constraints. The preliminary numerical experiments on some small benchmark test problems demonstrate our theoretical results.

math.OC↗

arXiv2021

Equipping Barzilai-Borwein method with two dimensional quadratic termination property

A novel gradient stepsize is derived at the motivation of equipping the Barzilai-Borwein (BB) method with two dimensional quadratic termination property. A remarkable feature of the novel stepsize is that its computation only depends on the BB stepsizes in previous iterations and does not require any exact line search or the Hessian, and hence it can easily be extended for nonlinear optimization. By adaptively taking long BB steps and some short steps associated with the new stepsize, we develop an efficient gradient method for quadratic optimization and general unconstrained optimization and extend it to solve extreme eigenvalues problems. The proposed method is further extended for box-constrained optimization and singly linearly box-constrained optimization by incorporating gradient projection techniques. Numerical experiments demonstrate that the proposed method outperforms the most successful gradient methods in the literature.

math.OC↗

arXiv2020

A variable metric mini-batch proximal stochastic recursive gradient algorithm with diagonal Barzilai-Borwein stepsize

Variable metric proximal gradient methods with different metric selections have been widely used in composite optimization. Combining the Barzilai-Borwein (BB) method with a diagonal selection strategy for the metric, the diagonal BB stepsize can keep low per-step computation cost as the scalar BB stepsize and better capture the local geometry of the problem. In this paper, we propose a variable metric mini-batch proximal stochastic recursive gradient algorithm VM-mSRGBB, which updates the metric using a new diagonal BB stepsize. The linear convergence of VM-mSRGBB is established for strongly convex, non-strongly convex and convex functions. Numerical experiments on standard data sets show that VM-mSRGBB is better than or comparable to some variance reduced stochastic gradient methods with best-tuned scalar stepsizes or BB stepsizes. Furthermore, the performance of VM-mSRGBB is superior to some advanced mini-batch proximal stochastic gradient methods.

math.OC↗

arXiv2020

On the acceleration of the Barzilai-Borwein method

The Barzilai-Borwein (BB) gradient method is efficient for solving large-scale unconstrained problems to the modest accuracy and has a great advantage of being easily extended to solve a wide class of constrained optimization problems. In this paper, we propose a new stepsize to accelerate the BB method by requiring finite termination for minimizing two-dimensional strongly convex quadratic function. Combing with this new stepsize, we develop gradient methods which adaptively take the nonmonotone BB stepsizes and certain monotone stepsizes for minimizing general strongly convex quadratic function. Furthermore, by incorporating nonmonotone line searches and gradient projection techniques, we extend these new gradient methods to solve general smooth unconstrained and bound constrained optimization. Extensive numerical experiments show that our strategies of properly inserting monotone gradient steps into the nonmonotone BB method could significantly improve its performance and the new resulted methods can outperform the most successful gradient decent methods developed in the recent literature.

math.OC↗

arXiv2019

On the asymptotic convergence and acceleration of gradient methods

We consider the asymptotic behavior of a family of gradient methods, which include the steepest descent and minimal gradient methods as special instances. It is proved that each method in the family will asymptotically zigzag between two directions. Asymptotic convergence results of the objective value, gradient norm, and stepsize are presented as well. To accelerate the family of gradient methods, we further exploit spectral properties of stepsizes to break the zigzagging pattern. In particular, a new stepsize is derived by imposing finite termination on minimizing two-dimensional strictly convex quadratic function. It is shown that, for the general quadratic function, the proposed stepsize asymptotically converges to the reciprocal of the largest eigenvalue of the Hessian. Furthermore, based on this spectral property, we propose a periodic gradient method by incorporating the Barzilai-Borwein method. Numerical comparisons with some recent successful gradient methods show that our new method is very promising.

math.OC↗

arXiv2019

Gradient methods exploiting spectral properties

We propose a new stepsize for the gradient method. It is shown that this new stepsize will converge to the reciprocal of the largest eigenvalue of the Hessian, when Dai-Yang's asymptotic optimal gradient method (Computational Optimization and Applications, 2006, 33(1): 73-88) is applied for minimizing quadratic objective functions. Based on this spectral property, we develop a monotone gradient method that takes a certain number of steps using the asymptotically optimal stepsize by Dai and Yang, and then follows by some short steps associated with this new stepsize. By employing one step retard of the asymptotic optimal stepsize, a nonmonotone variant of this method is also proposed. Under mild conditions, $R$-linear convergence of the proposed methods is established for minimizing quadratic functions. In addition, by combining gradient projection techniques and adaptive nonmonotone line search, we further extend those methods for general bound constrained optimization. Two variants of gradient projection methods combining with the Barzilai-Borwein stepsizes are also proposed. Our numerical experiments on both quadratic and bound constrained optimization indicate that the new proposed strategies and methods are very effective.

math.OC↗

arXiv2018

A family of spectral gradient methods for optimization

We propose a family of spectral gradient methods, whose stepsize is determined by a convex combination of the long Barzilai-Borwein (BB) stepsize and the short BB stepsize. Each member of the family is shown to share certain quasi-Newton property in the sense of least squares. The family also includes some other gradient methods as its special cases. We prove that the family of methods is $R$-superlinearly convergent for two-dimensional strictly convex quadratics. Moreover, the family is $R$-linearly convergent in the any-dimensional case. Numerical results of the family with different settings are presented, which demonstrate that the proposed family is promising.

math.OC↗

arXiv2018

A primal-dual interior-point relaxation method for nonlinear programs

We prove that the classic logarithmic barrier problem is equivalent to a particular logarithmic barrier positive relaxation problem with barrier and scaling parameters. Based on the equivalence, a line-search primal-dual interior-point relaxation method for nonlinear programs is presented. Our method does not require any primal or dual iterates to be interior-points, which is prominently different from the existing interior-point methods in the literature. A new logarithmic barrier penalty function dependent on both primal and dual variables is used to prompt the global convergence of the method, where the penalty parameter is updated adaptively. Without assuming any regularity condition, it is proved that our method will terminate at an approximate KKT point of the original problem provided the barrier parameter tends zero. Otherwise, either an approximate infeasible stationary point or an approximate singular stationary point of the original problem will be found. Some preliminary numerical results are reported, including the results for a well-posed problem for which many line-search interior-point methods were demonstrated not to be globally convergent, a feasible problem for which the LICQ and the MFCQ fail to hold at the solution and an infeasible problem, and for some standard test problems of the CUTE collection. These results show that our algorithm is not only efficient for well-posed feasible problems, but also is applicable for some ill-posed feasible problems and some even infeasible problems.

math.OC↗

arXiv2018

A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs

With the help of a logarithmic barrier augmented Lagrangian function, we can obtain closed-form solutions of slack variables of logarithmic-barrier problems of nonlinear programs. As a result, a two-parameter primal-dual nonlinear system is proposed, which corresponds to the Karush-Kuhn-Tucker point and the infeasible stationary point of nonlinear programs, respectively, as one of two parameters vanishes. Based on this distinctive system, we present a primal-dual interior-point method capable of rapidly detecting infeasibility of nonlinear programs. The method generates interior-point iterates without truncation of the step. It is proved that our method converges to a Karush-Kuhn-Tucker point of the original problem as the barrier parameter tends to zero. Otherwise, the scaling parameter tends to zero, and the method converges to either an infeasible stationary point or a singular stationary point of the original problem. Moreover, our method has the capability to rapidly detect the infeasibility of the problem. Under suitable conditions, not only the method can be superlinearly or quadratically convergent to the Karush-Kuhn-Tucker point as the original problem is feasible, but also it can be superlinearly or quadratically convergent to the infeasible stationary point when a problem is infeasible. Preliminary numerical results show that the method is efficient in solving some simple but hard problems and some standard test problems from the CUTE collection, where the superlinear convergence is demonstrated when we solve two infeasible problems and one well-posed feasible counterexample presented in the literature.

math.OC↗