Interior-point Optimization Algorithm Research Articles

Complexity analysis and numerical implementation of a new interior-point algorithm for semidefinite optimization

We generalize Zhang and Xu's (2011) [22] interior point algorithm for linear optimization to semidefinite optimization problems in order to define a new search direction. The symmetrization of the search direction is based on the full Nesterov-Todd scaling scheme. Moreover, we show that the obtained algorithm solves the studied problem in polynomial time and that the short-step algorithm has the best-known iteration bound, namely O(nlog⁡nε)-iterations. Finally, we report a comparative numerical study to show the efficiency of our proposed algorithm.

Two-factor Rough Bergomi Model: American Call Option Pricing and Calibration by Interior Point Optimization Algorithm

Two-factor Rough Bergomi Model: American Call Option Pricing and Calibration by Interior Point Optimization Algorithm

Large-step algorithm for linear complementarity problem with new search direction

Recently, [Darvay Zs, Takács PR: Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood. Cent Eur J Oper Res. 2018;26(3):551–563] introduced a new large-step interior-point algorithm (IPA) for linear optimization (LO). The purpose of this paper is to extend this method to the class of P ∗ ( κ ) -linear complementarity problems (LCPs). First, we propose a new wide neighbourhood of the central path for LCPs, which differs from the one presented by [Ai W, Zhang S: An O ( n L ) iteration primal–dual path-following method, based on wide neighbourhoods and large updates, for monotone LCP. SIAM J Optim. 2005;16(2):400–417]. Our method uses new search directions obtained by an algebraically equivalent transformation of the system which defines the central path. We present a large-step algorithm, but we demonstrate that it has the same iteration complexity as the best short-step algorithms. Finally, we give some numerical results to prove the efficiency of the new method.

A long-step second-order corrector interior point algorithm for linear optimization based on the algebraic equivalent transformation

In this paper, we present a new large-step primal-dual second-order corrector interior-point algorithm for linear optimization. This algorithm is based on the Nagy-Varga neighborhood of the central path, which uses the algebraic equivalent transformation strategy of Darvay et al. to determine the search directions. In addition, our method uses a second-order corrector direction in each iteration. The iterates remain in the proposed neighborhood by taking the largest possible step lengths along the search directions. We show that the new long-step algorithm is convergent and has the same complexity as the best short-step algorithms for linear optimization. To the best of our knowledge this is the first primal-dual second-order corrector interior-point algorithm that uses a different neighborhood from those available in the literature. Finally, numerical experiments show that the proposed algorithm is efficient and competitive.

Automated Regularized Deconvolution for Eliminating Extra-Column Effects in Fast High-Efficiency Separations.

With the introduction of ultrahigh efficiency columns and fast separations, the need to eliminate peak deformation contributed by the instrument must be effectively solved. Herein, we develop a robust framework to automate deconvolution and minimize its artifacts, such as negative dips, wild noise oscillations, and ringing, by combining regularized deconvolution and Perona-Malik (PM) anisotropic diffusion methods. A asymmetric generalized normal (AGN) function is proposed to model the instrumental response for the first time. With no-column data at various flow rates, the interior point optimization algorithm extracts the parameters describing instrumental distortion. The column-only chromatogram was reconstructed using the Tikhonov regularization technique with minimal instrumental distortion. For illustration, four different chromatography systems are used in fast chiral and achiral separations with 2.1 and 4.6 mm i.d. columns. Ordinary HPLC data can approach highly optimized UHPLC data. Similarly, in fast HPLC-circular dichroism (CD) detection, 8000 plates were gained for a fast chiral separation. Moment analysis of deconvolved peaks confirms correction of the center of mass, variance, skew, and kurtosis. This approach can be easily integrated and used with virtually any separation and detection system to provide enhanced analytical data.

Primal–dual interior-point algorithm for symmetric model predictive control

This paper presents a primal–dual interior-point (pdip) optimization algorithm for solving extreme-scale model predictive control (mpc) problems with linear dynamics, polytopic constraints, and quadratic/linear costs which are all invariant under the symmetric-group. We show that exploiting symmetry can reduce the computational and memory burden of extreme-scale or fast-paced applications of mpc. Our algorithm transforms the original inputs, states, and constraints of the mpc problem into a symmetric domain. The premise of our algorithm is that the numerical linear algebra used to solve the optimization problem has lower computational and memory complexity in the transformed domain. We demonstrate our algorithm for a heating, ventilation, and air-conditioning (hvac) numerical example. We show that, for our largest hvac control problem, our symmetry exploiting the pdip algorithm reduces the computation-time from minutes to seconds in comparison with the baseline pdip algorithm. Furthermore, we show that the presented symmetry exploiting pdip algorithm outperforms a state-of-the-art symmetry exploiting optimization algorithm.

Web Bend-Buckling of Steel Plate Girders Reinforced by Two Longitudinal Stiffeners with Various Cross-Section Shapes

This work performs an investigation into the optimal position of two longitudinal stiffeners with different cross-section shapes such as open section (L-shaped and T-shaped) and closed section (rectangular and triangular shapes) shapes of stiffened plate girders under bending loading through an optimization procedure using a gradient-based interior point (IP) optimization algorithm. The stiffener optimum locations are found by maximizing the bend-buckling coefficient, kb, generated from eigenvalue buckling analyses in Abaqus. The optimization procedure efficiently combines the finite element method and the IP optimization algorithm and is implemented using the Abaqus2Matlab toolbox which allows for the transfer of data between Matlab and Abaqus and vice versa. It is found that the proposed methodology can lead to the optimum design of the steel plate girder for all stiffener cross-section types with an acceptable accuracy and a reduced computational effort. Based on the optimization results, the optimum positions of two longitudinal stiffeners with various cross-section shapes are presented for the first time. It is reported that the optimum locations of two longitudinal stiffeners with open cross-section shapes (T- and L-shaped) are similar to that of flat cross-section, while the optimum positions of two longitudinal stiffeners with closed cross-section types (rectangular and triangular sections) are slightly different. One of the main findings of this study is that the bend-buckling coefficient of the stiffened girder having stiffeners with triangular cross-section shape is highest while that with flat cross-section shape is lowest among all considered stiffener types and this latter case has minimum requirement regarding the web thickness.

Complexity analysis of an interior-point algorithm for linear optimization based on a new parametric kernel function with a double barrier term

<p style='text-indent:20px;'>Kernel functions play an important role in the complexity analysis of the interior point methods (IPMs) for linear optimization (LO). In this paper, an interior-point algorithm for LO based on a new parametric kernel function is proposed. By means of some simple analysis tools, we prove that the primal-dual interior-point algorithm for solving LO problems meets <inline-formula><tex-math id="M1">\begin{document}$ O\left(\sqrt{n} \log(n) \log(\frac{n}{\varepsilon}) \right) $\end{document}</tex-math></inline-formula>, iteration complexity bound for large-update methods with the special choice of its parameters.</p>

A wide neighbourhood primal-dual second-order corrector interior point algorithm for semidefinite optimization

In this paper, we proposed a new primal-dual second-order corrector interior-point algorithm for semidefinite optimization. The algorithm is based on Darvay–Takács neighbourhood of the central path. In the new algorithm, the search directions are determined by the Darvay–Takács's direction and a second-order corrector direction in each iteration. The iteration complexity bound is for the Nesterov–Todd scaling direction, which coincides with the best-known complexity results for semidefinite optimization. Finally, numerical experiments show that the proposed algorithm is promising.

Nonlinear homotopy interior-point algorithm for 6-DoF powered landing guidance

The six-degree-of-freedom (6-DoF) powered landing guidance problem usually is difficult to solve for several reasons. First, this problem coupled with low-frequency translational and high-frequency rotational equations of motion is naturally large-scale. Second, the constraints enforced into the problem are highly coupled, nonlinear and non-convex, which makes the problem quite sensitive to the initial value guesses. Third, the initial position, velocity and attitude of the landing vehicle is quite difficult to forecast due to the complex endo-atmospheric environment, thus the traditional guidance methods are difficult to obtain a predetermined trajectory. In this paper, for the first time we proposed a nonlinear homotopy interior-point optimization algorithm (NHOPT) for solving the 6-DoF powered landing guidance problem. Different from general external homotopy strategies, we embed the concept of homotopy methods into the barrier line-search interior-point method. Using vertical landing without position and velocity offset, it is easy to provide initial guesses, which we define as the initial homotopy subproblem. Then a class of related homotopy subproblems is solved along a homotopy path using the algorithm until the original problem is converged. Numerical results demonstrate that the algorithm has high performance in terms of convergence and computational efficiency.

Optimum location of a single longitudinal stiffener with various cross-section shapes of steel plate girders under bending loading

This paper aims at investigating the optimum location of a single longitudinal stiffener having various cross-section types of steel plate girders subjected to bending loading. The optimum location is found using a gradient-based Interior-Point (IP) optimization algorithm. The optimization procedure involves a linear elastic buckling analysis with the finite element method, which is coupled with the aforementioned IP algorithm and used to calculate the buckling coefficient, which is maximized. Based on the optimization results, the effect of the various cross-section types of the stiffener and the web slenderness ratio on the optimum stiffener position is investigated. It is found that closed cross-sections of the stiffener and larger web slenderness ratios lead to higher critical buckling coefficients.

A polynomial time infeasible interior-point arc-search algorithm for convex optimization

This paper proposes an infeasible interior-point algorithm for the convex optimization problem using arc-search techniques. The proposed algorithm simultaneously selects the centering parameter and the step size, aiming at optimizing the performance in every iteration. Analytic formulas for the arc-search are provided to make the arc-search method very efficient. The convergence of the algorithm is proved and a polynomial bound of the algorithm is established. The preliminary numerical test results indicate that the algorithm is efficient and effective

Optimization of the carbon nanotubes reinforcement in cement-based materials

During the last decades, composites had been successfully strengthened at the micrometer scale using different types of reinforcements, e.g., carbon or even glass fibers. Over the past years, the development of carbon nanotubes and graphene enabled the reinforcement of cement-based materials at the nanoscale. The produced nanocomposites have increased the mechanical properties as compared to non-reinforced materials and especially on the “effective” properties such as modulus of elasticity and Poisson’s ratio. In several cases and for compatibility issues, it is of imperative importance to “tailor” a nano-reinforced composite with specific material properties, such as the modulus of elasticity and the Poisson’s ratio. To this end, a methodology to identify the optimal material synthesis needs to be effectively addressed. The present article presents an optimization method to quantify the appropriate volume fraction of graphene nanostructures to design a cement-based nanocomposite with a target value of the effective modulus. The Interior Point Optimization (IPOpt) algorithm is used for the optimization iterations applied to the homogenization properties calculated by ANSA® pre-processor. A comparison was made between three different optimization cases, having different maximum number of iterations and a different objective value. The calculated effective modulus and volume fraction were correlated with the target experimental values.

Theoretical and numerical result for linear semidefinite programming based on a new kernel function

Kernel functions serve the central goal of creating new search directions for the primal-dual interiorpoint algorithm to solve linear optimization problems. A significantly improved primal-dual interior-point algorithm for linear optimization is presented based on a novel kernel function. We show a primal-dual interior-point technique for linear optimization based on a class of kernel functions that are eligible. This research presents a new efficient kernel function-based primal-dual IPM algorithm for semidefinite programming problems based on the Nesterov-Todd (NT) direction. With a new and simple technique, we propose a new kernel function to obtain an optimal solution of the perturbed problem (SDP)µ. We obtain the best-known complexity results,for smalland large-update namely O(pp+1/2p √nlog tr(X0S0)/ε) and O((pn)p+1/2p log trX0S0/ε) large update To prove the effectiveness of our proposed kernel function, we compare our numerical results with some alternatives presented by Touil et al. (2017).

An infeasible interior-point arc-search algorithm for nonlinear constrained optimization

In this paper, we propose an infeasible arc-search interior-point algorithm for solving nonlinear programming problems. Most algorithms based on interior-point methods are categorized as line search since they compute a next iterate on a straight line determined by a search direction which approximates the central path. The proposed arc-search interior-point algorithm uses an arc for the approximation. We discuss convergence properties of the proposed algorithm. We also conduct numerical experiments on the CUTEst benchmark problems and compare the performance of the proposed arc-search algorithm with that of a line-search algorithm. Numerical results indicate that the proposed arc-search algorithm reaches the optimal solution using fewer iterations but longer times than a line-search algorithm. A modification that leads to a faster arc-search algorithm is also discussed.

Topology optimization for blood flow considering a hemolysis model

In the past years, topology optimization has been studied for designing fluid flow devices, such as channels, valves and pumps, and also considering non-Newtonian fluid flows, such as blood. When considering blood flow devices, it is important to quantify and minimize the blood damage (given mainly by hemolysis). However, up to now, in topology optimization, hemolysis has been minimized in an indirect manner, by considering shear stress (or even energy dissipation) as the objective function to indirectly minimize hemolysis. This approach may give a general idea of where hemolysis may or may not appear, but the actual distribution of hemolysis can not be easily correlated. Therefore, a more direct measure may be better to evaluate hemolysis. The direct way to measure hemolysis is by considering the hemolysis index, which is given from a differential equation model (a “hemolysis model”). In this work, the hemolysis index computed though a hemolysis model is included in the topology optimization formulation. In order to illustrate this approach, the design of a 2D swirl flow device, which is based on an axisymmetric fluid flow with or without rotation around an axis, is considered. One relevant example in the field of blood flow devices is the design of blood pumps, which has been previously considered in topology optimization with the aim of indirectly reducing hemolysis. More specifically in terms of pump design for 2D swirl flow, the design of a Tesla-type blood pump is considered. A Tesla-type pump is a bladeless fluid flow device, in which the boundary layer effect is used for pumping the fluid. This principle of operation may lead to a smaller induction of blood damage. Together with the hemolysis index, the topology optimization is formulated by considering the relative energy dissipation for indirectly maximizing efficiency. The fluid is modeled considering a non-Newtonian fluid model, and the fluid flow is solved with the finite element method. In order to model the solid material to block the fluid flow, the traditional formulation of fluid topology optimization is augmented with the “Brinkman-Forchheimer model”. Also, an additional penalization is considered in the non-Newtonian viscosity. The optimization problem is solved with IPOPT (Interior Point Optimization algorithm).

A primal-dual interior-point algorithm for nonsymmetric exponential-cone optimization

A new primal-dual interior-point algorithm applicable to nonsymmetric conic optimization is proposed. It is a generalization of the famous algorithm suggested by Nesterov and Todd for the symmetric conic case, and uses primal-dual scalings for nonsymmetric cones proposed by Tunçel. We specialize Tunçel’s primal-dual scalings for the important case of 3 dimensional exponential-cones, resulting in a practical algorithm with good numerical performance, on level with standard symmetric cone (e.g., quadratic cone) algorithms. A significant contribution of the paper is a novel higher-order search direction, similar in spirit to a Mehrotra corrector for symmetric cone algorithms. To a large extent, the efficiency of our proposed algorithm can be attributed to this new corrector.

An algorithm for nonsymmetric conic optimization inspired by MOSEK

We analyse the scaling matrix, search direction, and neighbourhood used in MOSEK's algorithm for nonsymmetric conic optimization [J. Dahl and E.D. Andersen, A primal-dual interior-point algorithm for nonsymmetric exponential-cone optimization, preprint (2019)]. It is proven that these can be used to compute a near-optimal solution to the homogeneous self-dual model in polynomial time. This provides a theoretical foundation for MOSEK's nonsymmetric conic algorithm. The main steps in the analysis are sandwiching MOSEK's scaling matrix between the primal and dual barrier's Hessians, and using this information to carefully check all the neighbourhood conditions after a small, improving step is taken.

A New Predictor-corrector Infeasible Interior-point Algorithm for Linear Optimization in aWide Neighborhood

In this paper, we propose a Mizuno-Todd-Ye type predictor-corrector infeasible interior-point method for linear optimization based on a wide neighborhood of the central path. According to Ai-Zhang’s original idea, we use two directions of distinct and orthogonal corresponding to the negative and positive parts of the right side vector of the centering equation of the central path. In the predictor stage, the step size along the corresponded infeasible directions to the negative part is chosen. In the corrector stage by modifying the positive directions system a full-Newton step is removed. We show that, in addition to the predictor step, our method reduces the duality gap in the corrector step and this can be a prominent feature of our method. We prove that the iteration complexity of the new algorithm is 𝒪(n log ɛ−1), which coincides with the best known complexity result for infeasible interior-point methods, where ɛ > 0 is the required precision. Due to the positive direction new system, we improve the theoretical complexity bound for this kind of infeasible interior-point method [1] by a factor of n . Numerical results are also provided to demonstrate the performance of the proposed algorithm.

An arc-search infeasible interior-point method for semidefinite optimization with the negative infinity neighborhood

We present an arc-search infeasible interior-point algorithm for semidefinite optimization using the Nesterov-Todd search directions. The algorithm is based on the negative infinity neighborhood of the central path. The algorithm searches an e-approximate solution of the problem along the ellipsoidal approximation of the entire central path. The convergence analysis of the algorithm is presented and shows that the algorithm has the iteration complexity bound $\mathcal {O}\big (n^{3/2}\log {\varepsilon }^{-1}\big )$ . Here, n is the dimension of the problem and e is the required precision. The numerical results show that our algorithm is efficient and promising.