Full Nesterov-Todd Step Research Articles

Interior-point algorithm for symmetric cone horizontal linear complementarity problems based on a new class of algebraically equivalent transformations

AbstractWe generalize a primal-dual interior-point algorithm (IPA) proposed recently in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022) to $$P_*(\kappa )$$ P ∗ ( κ ) -horizontal linear complementarity problems (LCPs) over Cartesian product of symmetric cones. The algorithm is based on the algebraic equivalent transformation (AET) technique with a new class of AET functions. The new class is a modification of the class of AET functions proposed in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022) where only two conditions are used as opposed to three used in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022). Furthermore, the algorithm is a feasible algorithm that uses full Nesterov-Todd steps, hence, no calculation of step-size is necessary. Nevertheless, we prove that the proposed IPA has the iteration bound that matches the best known iteration bound for IPAs solving these types of problems.

A class of new search directions for full-NT step feasible interior point method in semidefinite optimization

In this paper, based on Darvay et al.’s strategy for linear optimization (LO) (Z. Darvay and P.R. Takács, Optim. Lett. 12 (2018) 1099–1116.), we extend Kheirfam et al.’s feasible primal-dual path-following interior point algorithm for LO (B. Kheirfam and A. Nasrollahi, Asian-Eur. J. Math. 1 (2020) 2050014.) to semidefinite optimization (SDO) problems in order to define a class of new search directions. The algorithm uses only full Nesterov-Todd (NT) step at each iteration to find an ε-approximated solution to SDO. Polynomial complexity of the proposed algorithm is established which is as good as the LO analogue. Finally, we present some numerical results to prove the efficiency of the proposed algorithm.

A Full Nesterov-Todd Step Infeasible Interior-point Method for Symmetric Optimization in the Wider Neighborhood of the Central Path.

In this paper, an improved Interior-Point Method (IPM) for solving symmetric optimization problems is presented. Symmetric optimization (SO) problems are linear optimization problems over symmetric cones. In particular, the method can be efficiently applied to an important instance of SO, a Controlled Tabular Adjustment (CTA) problem which is a method used for Statistical Disclosure Limitation (SDL) of tabular data. The presented method is a full Nesterov-Todd step infeasible IPM for SO. The algorithm converges to ε-approximate solution from any starting point whether feasible or infeasible. Each iteration consists of the feasibility step and several centering steps, however, the iterates are obtained in the wider neighborhood of the central path in comparison to the similar algorithms of this type which is the main improvement of the method. However, the currently best known iteration bound known for infeasible short-step methods is still achieved.

Full Nesterov-Todd step feasible interior-point algorithm for symmetric cone horizontal linear complementarity problem based on a positive-asymptotic barrier function

We present a feasible full step interior-point algorithm to solve the horizontal linear complementarity problem defined on a Cartesian product of symmetric cones, which is not based on a usual barrier function. The full steps are scaled utilizing the Nesterov-Todd (NT) scaling point. Our approach generates the search directions leading to the full-NT steps by algebraically transforming the centring equation of the system which defines the central trajectory using the induced barrier of a so-called positive-asymptotic kernel function. We establish the global convergence as well as a local quadratic rate of convergence of our proposed method. Finally, we demonstrate that our algorithm bears a complexity bound matching the best available one for the algorithms of its kind.

Simplified full Nesterov–Todd step infeasible interior-point algorithm for semidefinite optimization based on a kernel function

In this paper, we propose a new complexity analysis of the full Nesterov–Todd step infeasible interior-point algorithm for semidefinite optimization. With a specific feasibility step and the centering step induced by a well-known kernel function, the property of exponential convexity of the kernel function underlying the matrix barrier function is crucial in the analysis and enable us easily to estimate the proximity of iterates to center path. The analysis of the algorithm is simplified and the iteration bound obtained coincides with the currently best iteration bound for infeasible interior-point algorithm.

Complexity analysis of infeasible interior-point method for semidefinite optimization based on a new trigonometric kernel function

In this paper, a full Nesterov–Todd step infeasible interior-point method for solving semidefinite optimization problems based on a new kernel function is analyzed. In each iteration, the algorithm involves a feasibility step and several centrality steps. The centrality step is focused on Nesterov–Todd search directions, while we used a kernel function with trigonometric barrier term to induce the feasibility step. The complexity result coincides with the best-known iteration bound for infeasible interior-point methods.

New complexity analysis of full Nesterov-Todd step infeasible interior point method for second-order cone optimization

We present a full Nesterov-Todd (NT) step infeasible interior-point algorithm for second-order cone optimization based on a different way to calculate feasibility direction. In each iteration of the algorithm we use the largest possible barrier parameter value ?. Moreover, each main iteration of the algorithm consists of a feasibility step and a few centering steps. The feasibility step differs from the feasibility step of the other existing methods. We derive the complexity bound which coincides with the best known bound for infeasible interior point methods.

A full Nesterov-Todd step primal-dual path-following interior-point algorithm for semidefinite linear complementarity problems

A full Nesterov-Todd step primal-dual path-following interior-point algorithm for semidefinite linear complementarity problems

New complexity analysis of a full Nesterov–Todd step interior-point method for semidefinite optimization

In this paper, we propose a new primal-dual path-following interior-point method for semidefinite optimization based on a new reformulation of the nonlinear equation of the system which defines the central path. The proposed algorithm takes only full Nesterov and Todd steps and therefore no line-searches are needed for generating the new iterations. The convergence of the algorithm is established and the complexity result coincides with the best-known iteration bound for semidefinite optimization problems.

Erratum to: Improved Complexity Analysis of Full Nesterov-Todd Step Feasible Interior-Point Method for Symmetric Optimization

Erratum to: Improved Complexity Analysis of Full Nesterov-Todd Step Feasible Interior-Point Method for Symmetric Optimization

An infeasible full NT-step interior point method for circular optimization

In this paper, we design a primal-dual infeasible interior-point method for circular optimization that uses only full Nesterov-Todd steps. Each main iteration of the algorithm consisted of one so-called feasibility step. Furthermore, giving a complexity analysis of the algorithm, we derive the currently best-known iteration bound for infeasible interior-point methods.

An improved and modified infeasible interior-point method for symmetric optimization

In this paper an improved and modified version of full Nesterov–Todd step infeasible interior-point methods for symmetric optimization published in [A new infeasible interior-point method based on Darvay’s technique for symmetric optimization, Ann. Oper. Res. 211(1) (2013) 209–224; G. Gu, M. Zangiabadi and C. Roos, Full Nesterov–Todd step infeasible interior-point method for symmetric optimization, European J. Oper. Res. 214(3) (2011) 473–484; Simplified analysis of a full Nesterov–Todd step infeasible interior-point method for symmetric optimization, Asian-Eur. J. Math. 8(4) (2015) 1550071, 14 pp.] is considered. Each main iteration of our algorithm consisted of only a feasibility step, whereas in the earlier versions each iteration is composed of one feasibility step and several — at most three — centering steps. The algorithm finds an [Formula: see text]-solution of the underlying problem in polynomial-time and its iteration bound improves the earlier bounds factor from [Formula: see text] and [Formula: see text] to [Formula: see text]. Moreover, our method unifies the analysis for linear optimization, second-order cone optimization and semidefinite optimization.

An infeasible full-NT step interior point algorithm for CQSCO

In this paper, we propose a full Nesterov-Todd (NT) step infeasible interior-point algorithm for convex quadratic symmetric cone optimization based on Euclidean Jordan algebra. The algorithm uses only one feasibility step in each main iteration. The complexity result coincides with the best-known iteration bound for infeasible interior-point methods.

Simplified analysis of a full Nesterov–Todd step infeasible interior-point method for symmetric optimization

We give a simplified analysis and an improved iteration bound of a full Nesterov–Todd (NT) step infeasible interior-point method for solving symmetric optimization. This method shares the features as, it (i) requires strictly feasible iterates on the central path of a perturbed problem, (ii) uses the feasibility steps to find strictly feasible iterates for the next perturbed problem, (iii) uses the centering steps to obtain a strictly feasible iterate close enough to the central path of the new perturbed problem, and (iv) reduces the size of the residual vectors with the same speed as the duality gap. Furthermore, the complexity bound coincides with the currently best-known iteration bound for full NT step infeasible interior-point methods.

A New Infeasible Interior-Point Method Based on a Non-Coercive Kernel Function with Improved Centering Steps for Second-Order Cone Optimization

In this article, we present a new full Nesterov-Todd step infeasible interior-point method for second-order cone optimization based on a non-coercive kernel function. The main iteration consists of a so-called feasibility step and one centering step, whereas the earlier versions, in [4, 21], needed two additional centering steps. We use a kernel function to induce the feasibility step. The new algorithm reduces the searching steps in each iteration and tenders an interesting analysis for complexity bound.

A full step infeasible interior-point method for Cartesian $$P_{*}(\kappa )$$ P ∗ ( κ ) -SCLCP

In this paper, a new full Nesterov–Todd step infeasible interior-point method for Cartesian \(P_*(\kappa )\) linear complementarity problem over symmetric cone is considered. Our algorithm starts from a strictly feasible point of a perturbed problem, after a full Nesterov–Todd step for the new perturbed problem the obtained strictly feasible iterate is close to the central path of it, where closeness is measured by some merit function. Furthermore, the complexity bound of the algorithm is the best available for infeasible interior-point methods.

Improved Complexity Analysis of Full Nesterov–Todd Step Interior-Point Methods for Semidefinite Optimization

In this paper, we present an improved convergence analysis of full Nesterov---Todd step feasible interior-point method for semidefinite optimization, and extend it to the infeasible case. This improvement due to a sharper quadratic convergence result, which generalizes a known result in linear optimization and leads to a slightly wider neighborhood for the iterates in the feasible algorithm and for the feasibility steps in the infeasible algorithm. For both versions of the full Nesterov---Todd step interior-point methods, we derive the same order of the iteration bounds as the ones obtained in linear optimization case.

An interior-point method for the Cartesian P*(k)-linear complementarity problem over symmetric cones

A novel primal-dual path-following interior-point algorithm for the Cartesian P*(k)-linear complementarity problem over symmetric cones is presented. The algorithm is based on a reformulation of the central path for finding the search directions. For a full Nesterov-Todd step feasible interior-point algorithm based on the new search directions, the complexity bound of the algorithm with small-update approach is the best-available bound.

A Full Nesterov-Todd Step Feasible Weighted Primal-Dual Interior-Point Algorithm for Symmetric Optimization

In this paper a weighted short-step primal-dual interior-point algorithm for linear optimization over symmetric cones is proposed that uses new search directions. The algorithm uses at each interior-point iteration a full Nesterov-Todd step and the strategy of the central path to obtain a solution of symmetric optimization. We establish the iteration bound for the algorithm, which matches the currently best-known iteration bound for these methods, and prove that the algorithm is quadratically convergent.

A new full Nesterov–Todd step feasible interior-point method for convex quadratic symmetric cone optimization

In this paper, we generalize the classical primal–dual logarithmic barrier method for linear optimization to convex quadratic optimization over symmetric cone by using Euclidean Jordan algebras. The symmetrization of the search directions used in this paper is based on the Nesterov–Todd scaling scheme, and only full Nesterov–Todd step is used at each iteration. We derive the iteration bound that matches the currently best known iteration bound for small-update methods, namely, O(rlogrε). Some preliminary numerical results are provided to demonstrate the computational performance of the proposed algorithm.