Interior Point Optimization of Low-Thrust Spacecraft Trajectories

In the last two decades, the mathematical programming community has witnessed some spectacular advances in interior point methods and robust optimization. These advances have recently started to significantly impact various fields of applied sciences and engineering where computational efficiency is essential. This paper focuses on two such fields: digital signal processing and communication. In the past, the widely used optimization methods in both fields had been the gradient descent or least squares methods, both of which are known to suffer from the usual headaches of stepsize selection, algorithm initialization and local minima. With the recent advances in conic and robust optimization, the opportunity is ripe to use the newly developed interior point optimization techniques and highly efficient software tools to help advance the fields of signal processing and digital communication. This paper surveys recent successes of applying interior point and robust optimization to solve some core problems in these two fields. The successful applications considered in this paper include adaptive filtering, robust beamforming, design and analysis of multi-user communication system, channel equalization, decoding and detection. Throughout, our emphasis is on how to exploit the hidden convexity, convex reformulation of semi-infinite constraints, analysis of convergence, complexity and performance, as well as efficient practical implementation.

The Karmarkar interior point algorithm has made it possible to solve large-scale decision problems that previously could not be solved in reasonable time, or were too large to be solved at all. In this paper, the authors present the results obtained from using the KORBX Advanced Mathematical Programming System (KMPS), which uses Karmarkar's interior point algorithm, to solve a number of linear optimization problems arising from a long-term fuel planning problem. Comparison with the results obtained using software based on the simplex method demonstrates the drastic improvements in solution time for the interior point method, especially with increase in problem sizes. This confirms earlier comparisons of interior point and simplex methods. The paper includes preliminary ideas and results on ways to combine the interior point and simplex methods in order to benefit from the superior speed performance of the former, and the warm-start and hot-start capabilities of the latter methods. >

The solution of nonlinear optimal power flow (OPF) problems by interior-point (IP) and noninterior-point (NIP) methods for nonlinear programming (NLP) is described. The mathematical development of both algorithms is based on a standard form NLP problem. A distinctive feature of the NIP method is that, in contrast to IP methods, it can start from arbitrary points. Test results are reported.

Crystal plasticity simulations offer insights into the anisotropic deformation of polycrystalline materials such as metals and alloys. However, rate‐independent crystal plasticity models encounter the Taylor ambiguity, where the active slip systems and plastic slip magnitudes are not uniquely defined, posing well‐known numerical challenges [1]. Interior point methods, which smooth the problem via a barrier term, have recently emerged as a promising strategy for both small‐strain [2, 3] and large‐strain [4, 5] crystal plasticity frameworks. This contribution presents a finite‐strain crystal plasticity model based on an interior point [2, 3]. Distinctly from existing large‐strain IPM implementations, our approach incorporates sequential updates for slip system rotations, which are held constant during intermediate interior point iterations, significantly enhancing algorithmic robustness. The method is evaluated at the material point level for several test cases, and its predictions are shown to be consistent with established results from various crystal plasticity algorithms in the literature, such as those based on an augmented Lagrange method.

Since conventional optimal power flow (OPF) algorithms are limited and too slow for operational planning, power system operation and planning engineers have been seeking efficient OPF algorithms for three decades. The interior point method is one of the most efficient algorithms. In this paper, an improved quadratic interior point (IQIP) method is used to solve comprehensive OPF problems with a variety of objective functions, including economic dispatch, VAr planning and loss minimization. The IQIP method features a general starting point (rather than selected good point as in the general interior point method) and fast convergence. Consequently, the OPF package described in this paper offers huge improvements in speed, accuracy and convergence in solving multi-objective and multi-constraint optimization problems. It will be a very useful tool for the power industry.

This paper describes a recently developed method for the determination of optimal, e g minimum-fuel or minimum-time, trajectories for spacecraft. The method will be introduced after a brief discussion of methods previously used by researchers for trajectory optimization. However, the objective here is not to discuss all such methods or provide a comprehensive description of them but rather to provide sufficient information regarding extant methods that what is novel and beneficial about the solution method we have developed may be recognized. Several example optimal trajectories for spacecraft using lowthrust propulsion will be presented.KeywordsOptimal Control ProblemOptimal TrajectoryPrime VectorAdjoint VariableSystem Differential EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this chapter, an implementation of an interior point filter line-search algorithm for large-scale nonlinear programming proposed by Wachter and Biegler (2005a, b) is presented. As we know, to allow convergence from poor starting points and to enforce progress to the solution, interior point methods both in trust region and in line-search frameworks with exact penalty merit function have been developed. For example, KNITRO uses the l 1 exact penalty function (Byrd, Hribar, & Nocedal, 1999; Byrd, Gilbert, & Nocedal, 2000). On the other hand, Fletcher and Leyffer (2002) proposed filter methods as an alternative to merit functions, as a tool for global convergence guarantee in algorithms for nonlinear optimization. The idea of filter is that trial points are accepted if they improve the objective function value or improve the constraint violation instead of a combination of these two measures defined by the merit function. Even if the filter methods include different heuristics, they have been adapted to barrier methods in a number of ways. For example, Ulbrich et al. (2004) considered a trust-region filter method and accept the trial step on the basis of the norm of the optimality conditions. Benson et al. (2002a) proposed several heuristics using the idea of filter methods, for which improved efficiency is reported compared to their previous merit function approach. The global convergence of an interior point algorithm with a filter line search was analyzed by Wachter and Biegler (2001). Here, the assumptions made for the analysis of the global convergence are less restrictive than those made for line-search interior point methods for nonlinear programming developed, for example, by El-Bakry et al. (1996), Yamashita (1998), or Tits et al. (2003).

Karmarkar's interior point method as a computation method for solving linear programming (LP) has attracted interest in the operation research community, due to its efficiency, reliability, and accuracy. This paper presents an extended quadratic interior point (EQIP) method, based on improvement of initial condition for solving both linear and quadratic programming problems, to solve power system optimization problem (PSOP), such as economic dispatch (ED) and VAr planning (VP) problems. The EQIP method is able to accommodate the nonlinearity in objectives and constraints. The scheme is demonstrated on several IEEE standard systems and is capable of achieving fast convergence and improvement in computational speed over an existing efficient Simplex, such as the MINOS code. The number of iterations during the computation is relatively insensitive to numbers of controls and constraints. Moreover, the proposed EQIP method guarantees a global optimum within the interior feasible region. >

Industrial energy systems are increasingly required to reduce operating costs and CO2 emissions while integrating variable renewable energy sources. Managing these objectives under uncertainty requires advanced optimization strategies capable of delivering reliable and real-time decisions. To address these challenges, this study focuses on the short-term operational planning of an industrial energy supply system using the rolling horizon approach (RHA). The RHA offers an effective framework to handle uncertainties by repeatedly updating forecasts and re-optimizing over a moving time window, thereby enabling adaptive and responsive energy management. To solve the resulting nonlinear and constrained optimization problem at each RHA iteration, we propose a novel hybrid algorithm that combines Bayesian optimization (BO) with the Interior Point OPTimizer (IPOPT). While global deterministic and stochastic optimization methods are frequently used in practice, they often suffer from high computational costs and slow convergence, particularly when applied to large-scale, nonlinear problems with complex constraints. To overcome these limitations, we employ the BO–IPOPT, integrating the global search capabilities of BO with the efficient local convergence and constraint fulfillment of the IPOPT. Applied to a large-scale real-world case study of a food and cosmetic industry in Germany, the proposed BO–IPOPT method outperformed state-of-the-art solvers in both solution quality and robustness, achieving up to 97.25%-better objective function values at the same CPU time. Additionally, the influence of key parameters, such as forecast uncertainty, optimization horizon length, and computational effort per RHA iteration, was analyzed to assess their impact on system performance and decision quality.

An approach that uses interior point methods is presented for teaching optimization theory to electrical and computer engineers. The features allow students to look at design projects such as VLSI circuit layout and simplify to solving linear systems of equations. The simplification to solve systems of symmetric positive-definite equations allows this material to be taught as part of a numerical methods course for electrical engineers. A simple junior level project is described for teaching interior point optimization and solutions of linear systems of equations. Additional material that would be suitable for senior or graduate-level courses on this topic is also suggested. >

This work investigated the computational efficiency of primal-dual interior-point methods for nonlinear convex optimization by refining both the underlying kernel functions and the barrier parameter update mechanisms. We introduced a unified parametric class of self-regular kernels that generalizes several established barrier families while maintaining optimal theoretical iteration complexity. To bridge the gap between theoretical convergence and practical performance, we proposed an adaptive update rule for the barrier parameter and evaluated various heuristics for its dynamic selection. Extensive numerical testing on a diverse benchmark suite demonstrated that the proposed framework significantly outperforms the Interior Point OPTimizer (IPOPT) solver while maintaining high numerical accuracy and minimal stationarity residuals. Moreover, the framework exhibited robust performance even on nonconvex problems, highlighting its practical versatility beyond the theoretical convex setting.

Considering the soft constraint characteristics of voltage constraints, the Interior-Point Filter Algorithm is applied to solve the formulation of fuzzy model for the power system reactive power optimization with a large number of equality and inequality constraints. Based on the primal-dual interior-point algorithm, the algorithm maintains an updating “filter” at each iteration in order to decide whether to admit correction of iteration point which can avoid effectively oscillation due to the conflict between the decrease of objective function and the satisfaction of constraints and ensure the global convergence. Moreover, the “filter” improves computational efficiency because it filters the unnecessary iteration points. The calculation results of a practical power system indicate that the algorithm can effectively deal with the large number of inequality constraints of the fuzzy model of reactive power optimization and satisfy the requirement of online calculation which realizes to decrease the network loss and maintain specified margins of voltage.

An interior-point algorithm for linear optimization based on a new barrier function

This paper proposes a nonlinear optimization interior point (IP) method for the determination of maximum loadability in a power system. Details of the implementation of pure primal-dual and predictor-corrector primal-dual IP algorithms are presented. It is shown that most of the computational effort of the algorithm is taken by the formation and factorization of the augmented Hessian matrix of the IP algorithm. The size of this matrix can be as large as ten times the number of buses in the system. Comparisons of the two IP implementations with large scale power systems with as many as 4000 buses are presented. It is shown that the IP algorithm constitutes an effective method for the determination of the maximum loadability in a power system.

Validation of measurement data using an interior point SQP