Wide Neighborhood Research Articles

An Arc Search Infeasible Interior-Point Algorithm for Symmetric Optimization Using a New Wide Neighborhood

In this paper, we propose an infeasible interior-point algorithm for symmetric optimization problems using a new wide neighborhood and estimating the central path by an ellipse. In contrast of most interior-point algorithms for symmetric optimization which search an $\varepsilon$ -optimal solution of the problem in a small neighborhood of the central path, our algorithm searches for optimizers in a new wide neighborhood of the ellipsoidal approximation of central path. The convergence analysis of the algorithm is shown and it is proved that the iteration bound of the algorithm is $O ( r\log\varepsilon^{-1} ) $ which improves the complexity bound of the recent proposed algorithm by Liu et al. (J. Optim. Theory Appl., 2013, https://doi.org/10.1007/s10957-013-0303-y ) for symmetric optimization by the factor $r^{\frac{1}{2}}$ and matches the currently best-known iteration bound for infeasible interior-point methods.

An Arc Search Interior-Point Algorithm for Monotone Linear Complementarity Problems over Symmetric Cones

An arc search interior-point algorithm for monotone symmetric cone linear complementarity problem is presented. The algorithm estimates the central path by an ellipse and follows an ellipsoidal approximation of the central path to reach an "-approximate solution of the problem in a wide neighborhood of the central path. The convergence analysis of the algorithm is derived. Furthermore, we prove that the algorithm has the complexity bound O ( p rL) using Nesterov-Todd search direction and O (rL) by the xs and sx search directions. The obtained iteration complexities coincide with the best-known ones obtained by any proposed interior- point algorithm for this class of mathematical problems.

Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood

The interior-point algorithms can be classified in multiple ways. One of these takes into consideration the length of the step. In this way, we can speak about large-step and short-step methods, that work in different neighbourhoods of the central path. The large-step algorithms work in a wide neighbourhood, while the short-step ones determine the new iterates that are in a smaller neighbourhood. In spite of the fact that the large-step algorithms are more efficient in practice, the theoretical complexity of the short-step ones is generally better. Ai and Zhang introduced a large-step interior-point method for linear complementarity problems using a wide neighbourhood of the central path, which has the same complexity as the best short-step methods. We present a new wide neighbourhood of the central path. We prove that the obtained large-step primal–dual interior-point method for linear programming has the same complexity as the best short-step algorithms.

Conditions for Error Bounds of Linear Complementarity Problems over Second-Order Cones with Pseudomonotonicity

We first discuss some properties of the solution set of a pseudomonotone second-order cone linear complementarity problem (SOCLCP), and then analyse the limiting behavior of a sequence of strictly feasible solutions within a new wide neighborhood of the central trajectory for the pseudomonotone SOCLCP under assumptions of strict complementarity. Based on this, we derive four different characterizations of an error bound for the pseudomonotone SOCLCP.

An interior-point algorithm for horizontal linear complementarity problems

This paper presents a wide-neighbourhood interior-point algorithm for P-horizontal linear complementarity problem. The convergence analysis is shown for the introduced wide neighbourhood of the central path by Ai and Zhang (2005) for monotone linear complementarity problem, and unifies the analysis for its constituent wide neighbourhoods. The Newton search directions are decomposed to the non-negative and non-positive parts, correspond to the parts of the right hand side. The achieved complexity bound is the same as the best obtained bound for the monotone linear complementarity problems, except that it is multiplied by a factor dependent on the handicap of the given problem.

A wide neighborhood primal-dual interior-point algorithm with arc-search for linear complementarity problems

A wide neighborhood primal-dual interior-point algorithm with arc-search for linear complementarity problems

Polynomial convergence of two higher order interior-point methods for $$P_*(\kappa )$$ P ∗ ( κ ) -LCP in a wide neighborhood of the central path

In this paper, we propose two interior-point methods for solving \(P_*(\kappa )\)-linear complementarity problems (\(P_*(\kappa )\)-LCPs): a high order large update path following method and a high order corrector–predictor method. Both algorithms generate sequences of iterates in the wide neighborhood \((\mathcal {N}_{2,\tau }^-(\alpha ))\) of the central path introduced by Ai and Zhang. The methods do not depend on the handicap \(\kappa \) of the problem so that they work for any \(P_*(\kappa )\)-LCP . They have \(O((1 +\kappa )\sqrt{n}L)\) iteration complexity, the best-known iteration complexity obtained so far by any interior-point method for solving \(P_*(\kappa )\)-LCP. The high order corrector–predictor algorithm is superlinearly convergent with Q-order \((m_p+1)\) for problems that admit a strict complementarity solution and \((m_p+1)/2\) for general problems, where \(m_p\) is the order of the predictor step.

A wide neighborhood infeasible-interior-point method with arc-search for -SCLCPs

In this paper, we propose an arc-search infeasible-interior-point method based on the wide neighbourhood for linear complementarity problems over symmetric cones with the Cartesian -property (-SCLCP). The algorithm searches the optimizers along the ellipses that approximate the central path. Moreover, we derive the complexity bound of the algorithm which coincides with the currently best known theoretical complexity bounds for the short step path-following algorithm. Some numerical results are provided to demonstrate the computational performance of the proposed algorithm.

A wide neighborhood arc-search interior-point algorithm for convex quadratic programming

In this paper, we propose an arc-search interior-point algorithm for convex quadratic programming with a wide neighborhood of the central path, which searches the optimizers along the ellipses that approximate the entire central path. The favorable polynomial complexity bound of the algorithm is obtained, namely O(nlog((x0)Ts0/e)) which is as good as the linear programming analogue. Finally, the numerical experiments show that the proposed algorithm is efficient.

A wide neighborhood primal-dual predictor-corrector interior-point method for symmetric cone optimization

We present a primal-dual predictor-corrector interior-point method for symmetric cone optimization. The proposed algorithm is based on the Nesterov-Todd search directions and a wide neighborhood, which is an even wider neighborhood than a given negative infinity neighborhood. At each iteration, the method computes two corrector directions in addition to the Ai and Zhang directions (SIAM J. Optim. 16, 400–417, 2005), in order to improve performance. Moreover, we derive the complexity bound of the wide neighborhood predictor-corrector interior-point method for symmetric cone optimization that coincides with the currently best known theoretical complexity bounds for the short step algorithm. Finally, some numerical experiments are provided to reveal the effectiveness of the proposed method.

A Wide Neighborhood Second-order Predictor-corrector Interior-point Algorithm for Semidefinite Optimization with Modified Corrector Directions

A Wide Neighborhood Second-order Predictor-corrector Interior-point Algorithm for Semidefinite Optimization with Modified Corrector Directions

A new infeasible-interior-point algorithm for linear programming over symmetric cones

In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N(τ1, τ2, η), for linear programming over symmetric cones. We treat the classical Newton direction as the sum of two other directions. We prove that if these two directions are equipped with different and appropriate step sizes, then the new algorithm has a polynomial convergence for the commutative class of search directions. In particular, the complexity bound is O(r1.5loge−1) for the Nesterov-Todd (NT) direction, and O(r2loge−1) for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and e > 0 is the required precision. If starting with a feasible point (x0, y0, s0) in N(τ1, τ2, η), the complexity bound is $$O\left( {\sqrt r \log {\varepsilon ^{ - 1}}} \right)$$ for the NT direction, and O(rloge−1) for the xs and sx directions. When the NT search direction is used, we get the best complexity bound of wide neighborhood interior-point algorithm for linear programming over symmetric cones.

Polynomial Convergence of Primal-Dual Path-Following Algorithms for Symmetric Cone Programming Based on Wide Neighborhoods and a New Class of Directions

This paper presents a class of primal-dual path-following interior-point algorithms for symmetric cone programming (SCP) based on wide neighborhoods and new directions with a parameter $$\theta $$ . When the parameter $$\theta =1$$ , the direction is exactly the classical Newton direction. When the parameter $$\theta $$ is independent of the rank of the associated Euclidean Jordan algebra, the algorithm terminates in at most $${\mathcal {O}}\left( \kappa r\log \varepsilon ^{-1}\right) $$ iterations, which coincides with the best known iteration bound for the classical wide neighborhood algorithms. When the parameter $$\theta =\sqrt{n/\beta \tau }$$ and Nesterov–Todd search direction is used, the algorithm has $${\mathcal {O}}\left( \sqrt{r}\log \varepsilon ^{-1}\right) $$ iteration complexity, the best iteration complexity obtained so far by any interior-point method for solving SCP. To our knowledge, this is the first time that a class of interior-point algorithms including the classical wide neighborhood path-following algorithm is proposed and analyzed over symmetric cone.

A Predictor-corrector Infeasible-interior-point Algorithm for Semidefinite Optimization in a Wide Neighborhood

A Predictor-corrector Infeasible-interior-point Algorithm for Semidefinite Optimization in a Wide Neighborhood

Two wide neighborhood interior-point methods for symmetric cone optimization

In this paper, we present two primal---dual interior-point algorithms for symmetric cone optimization problems. The algorithms produce a sequence of iterates in the wide neighborhood $$\mathcal {N}(\tau ,\,\beta )$$N(ź,β) of the central path. The convergence is shown for a commutative class of search directions, which includes the Nesterov---Todd direction and the xs and sx directions. We derive that these two path-following algorithms have $$\begin{aligned} \text{ O }\left( \sqrt{r\text{ cond }(G)}\log \varepsilon ^{-1}\right) , \text{ O }\left( \sqrt{r}\left( \text{ cond }(G)\right) ^{1/4}\log \varepsilon ^{-1}\right) \end{aligned}$$Orcond(G)logź-1,Orcond(G)1/4logź-1iteration complexity bounds, respectively. The obtained complexity bounds are the best result in regard to the iteration complexity bound in the context of the path-following methods for symmetric cone optimization. Numerical results show that the algorithms are efficient for this kind of problems.

On Superlinear and Convergence of a Path-Following Algorithm for Monotone Linear Complementarity Problems in a Wide Neighborhood

ABSTRACTWe propose a modified path-following algorithm in a wide neighborhood for monotone linear complementarity problems. The algorithm takes two steps in one iteration, called a fast step and a safe step, respectively. In the algorithm, if a fast step fails, a safe step will be taken. However, when iterations are sufficiently large, fast steps are taken all through. We prove that the algorithm has not only the complexity bound but also local superlinear convergence. Furthermore, assuming a strictly complementary solution exists, the sequence , generated by the algorithm, is convergent. Finally, the numerical results show the algorithm effective.

Primal-dual entropy-based interior-point algorithms for linear optimization

We propose a family of search directions based on primal-dual entropy in the context of interior-point methods for linear optimization. We show that by using entropy-based search directions in the predictor step of a predictor-corrector algorithm together with a homogeneous self-dual embedding, we can achieve the current best iteration complexity bound for linear optimization. Then, we focus on some wide neighborhood algorithms and show that in our family of entropy-based search directions, we can find the best search direction and step size combination by performing a plane search at each iteration. For this purpose, we propose a heuristic plane search algorithm as well as an exact one. Finally, we perform computational experiments to study the performance of entropy-based search directions in wide neighborhoods of the central path, with and without utilizing the plane search algorithms.

A primal–dual predictor–corrector interior-point method for symmetric cone programming with O(√r log ϵ−1) iteration complexity

ABSTRACTIn this paper,we propose a new predictor–corrector interior-point method for symmetric cone programming. This algorithm is based on a wide neighbourhood and the Nesterov–Todd direction. We prove that, besides the predictor steps, each corrector step also reduces the duality gap by a rate of , where r is the rank of the associated Euclidean Jordan algebras. In particular, the complexity bound is , where is a given tolerance. To our knowledge, this is the best complexity result obtained so far for interior-point methods with a wide neighbourhood over symmetric cones. The numerical results show that the proposed algorithm is effective.

The Fermi–Dirac distribution as a model of a thermodynamically ideal liquid. Phase transition of the first kind for neutral gases (corresponding to nonpolar molecules)

In this paper, we introduce the notions of enlarged number theory and of thermodynamically ideal liquid and calculate the temperature below which it appears. This temperature is T = 0.84Tc, where Tc is the critical temperature of a gas whose molecules are nonpolar. For such a gas, in a sufficiently wide neighborhood of the binodal, the isotherms of a gas and of a thermodynamically ideal liquid coincide with those of a van der Waals gas for the critical value of the compressibility factor Zc = 3/8. In this sense, for T ≤ 0.84Tc and the particular case Zc = 3/8, the developed theory is a generalization of the van der Waals model. A new phase transition of the second kind at the point of zero activity is described.

A new second-order corrector interior-point algorithm for P*(k)-LCP

In this paper, we propose a second-order corrector interior-point algorithm for solving P*(k)- linear complementarity problems. The method generates a sequence of iterates in a wide neighborhood of the central path introduced by Ai and Zhang. In each iteration, the method computes a corrector direction in addition to the Ai-Zhang direction, in an attempt to improve performance. The algorithm does not depend on the handicap k of the problem, so that it can be used for any P*(k)-linear complementarity problems. It is shown that the iteration complexity bound of the algorithm is O ((1+k)3 ? nL). Some numerical results are provided to illustrate the performance of the algorithm.