arXiv subjects

Detong Zhu

Publications and source records attributed to Detong Zhu.

4 recordsLinked to original sources

arXiv2026

A Second-Order Algorithm Based on Affine Scaling Interior-Point Methods for nonlinear Optimisation with bound constraints

The homogeneous second-order descent method (Zhang et al. 2025, Mathematics of Operations Research) was initially proposed for unconstrained optimisation problems. HSODM shows excellent performance with respect to the global complexity rate among a certain broad class of second-order methods. In this paper, we extend HSODM to solve nonlinear optimisation problems with bound constraints and propose a second-order algorithm based on affine scaling interior-point methods (SOBASIP). In each iteration, an appropriate affine matrix is introduced to construct an affine scaling subproblem based on the optimality conditions of the problem. To obtain a valid descent direction similar to HSODM, we utilise the homogenisation technique to transform the scaling subproblem into an Ordinary Homogeneous Model (OHM), which is essentially an eigenvalue problem that can be solved efficiently. The descent direction is constructed from the optimal solution to the OHM, and then backtracking line search is used to determine the new iteration point. Theoretical analysis establishes that SOBASIP achieves a global iteration complexity of $O(ε^{-3/2})$ for finding an $ε$-approximate second-order stationary point and converges locally at a superlinear rate under certain conditions. Numerical results demonstrate that the proposed method exhibits satisfactory performance.

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arXiv2026

A scalable sequential adaptive cubic regularization algorithm for optimization with general equality constraints

The scalable adaptive cubic regularization method ($\mathrm{ARC_{q}K}$: Dussault et al. in Math. Program. Ser. A 207(1-2): 191-225, 2024) has been recently proposed for unconstrained optimization. It has excellent convergence properties, well-defined complexity bounds, and promising numerical performance. In this paper, we extend $\mathrm{ARC_{q}K}$ to nonlinear optimization with general equality constraints and propose a scalable sequential adaptive cubic regularization algorithm named $\mathrm{SSARC_{q}K}$. In each iteration, we construct an ARC subproblem with linearized constraints inspired by sequential quadratic optimization methods. Next, a composite-step approach is used to decompose the trial step into the sum of a vertical step and a horizontal step. By means of the reduced-Hessian approach, we rewrite the linearly constrained ARC subproblem as a standard unconstrained ARC subproblem to compute the horizontal step. Analogous to $\mathrm{ARC_{q}K}$, we employ a CG-Lanczos procedure with shifts to solve ARC subproblems inexactly, thus bypassing any hard case consideration. This also avoids solving the subproblem multiple times for obtaining a new iterative point. We establish the global convergence of the inexact ARC method $\mathrm{SSARC_{q}K}$ to first-order critical points. Preliminary numerical tests and some comparison results are presented to illustrate the performance of $\mathrm{SSARC_{q}K}$.

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arXiv2025

A line search filter sequential adaptive cubic regularisation algorithm for nonlinearly constrained optimization

In this paper, a sequential adaptive regularization algorithm using cubics (ARC) is presented to solve nonlinear equality constrained optimization. It is motivated by the idea of handling constraints in sequential quadratic programming methods. In each iteration, we decompose the new step into the sum of the normal step and the tangential step by using composite step approaches. Using a projective matrix, we transform the constrained ARC subproblem into a standard ARC subproblem which generates the tangential step. After the new step is computed, we employ line search filter techniques to generate the next iteration point. Line search filter techniques enable the algorithm to avoid the difficulty of choosing an appropriate penalty parameter in merit functions and the possibility of solving ARC subproblem many times in one iteration in ARC framework. Global convergence is analyzed under some mild assumptions. Preliminary numerical results and comparison are reported.

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arXiv2024

An Adaptive Cubic Regularisation Algorithm Based on Interior-Point Methods for Optimization with General Inequality Constraints

Nonlinear constrained optimization has a wide range of practical applications. The interior-point method is considered to be one of the most powerful algorithms for solving nonlinear inequality constrained optimization. In this paper, we consider optimization with general inequality constraints and propose an Adaptive Regularisation algorithm using Cubics Based on Interior-Point methods (ARCBIP). For solving the barrier problem, we construct ARC subproblem with linearized constraints and the well-known fraction to the boundary rule that prevents slack variables from approaching their lower bounds prematurely. We employ a composite-step approach and reduced Hessian methods to deal with linearized constraints, where the trial step is decomposed into a normal step and a tangential step. They are obtained by solving two ARC subproblems approximately with the fraction to the boundary rule. Requirements on normal steps and tangential steps are given to ensure global convergence. To determine whether the trial step is accepted, we use exact penalty function as the merit function in ARC framework. The updating of the barrier parameter is implemented by adaptive strategies. Global convergence is analyzed under mild assumptions. Preliminary numerical experiments and some comparison results are reported.

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