Advanced economic dispatching control algorithm using dynamic unit model and interior point method

In this paper, we propose a theoretical framework of a predictor-corrector interior-point method for linear optimization based on the one-norm wide neighborhood of the central path, focusing on infeasible corrector steps. Here, we call the predictor-infeasible corrector algorithm. At each iteration, the proposed algorithm computes an infeasible corrector step in addition to the Ai-Zhang search directions and considers the Newton direction as a linear combination of these directions. We represent the complexity analysis of the algorithm and conclude that its iteration bound is $${\mathcal {O}}(n\log \varepsilon ^{-1})$$ . To our knowledge, this is the best complexity result up to now for infeasible interior-point methods based on these kinds of search directions. The complexity bound obtained here is the same as small neighborhood infeasible interior point algorithms.

This paper presents an adaptation of the Ant System method to find the optimal control input for blood glucose regulation using Model Predictive Control (MPC). The Ant System optimization method was implemented to solve a linear MPC problem and performance was compared with the interior point method for optimization. The Ant System was found to perform well for the linear MPC problem and has the advantage over the interior point method as it can extended for use with non-linear MPC problems.

It is known that predictor-corrector methods in a large neighborhood of the central path are among the most efficient interior-point methods (IPMs) for linear optimization (LO) problems. The best iteration bound based on the classical logarithmic barrier function is O(nlog (n/∊)). In this paper, we propose a family of self-regular proximity-based predictor-corrector (SRPC) IPMs for LO in a large neighborhood of the central path. In the predictor step, we use either an affine scaling or a self-regular direction; in the corrector step, we use always a self-regular direction. Our new algorithms use a special proximity function with different search directions and thus allows us to improve the so far best theoretical iteration complexity for a family of SRPC IPMs. An O\((\sqrt{n}{\exp} ((1 - q + {\log} n)/2) {\log} n {\log} (n/\epsilon))\) worst-case iteration bound with quadratic convergence is established, where q is the barrier degree of the SR proximity function.

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.

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. >

Recently the authors introduced the notions of self-regular functions and self-regular proximity functions and used them in the design and analysis of interior-point methods (IPMs) for linear and semidefinite optimization (LO and SDO). In this paper, we consider an extension of these concepts to second-order conic optimization (SOCO). This nontrivial extension requires the development of various new tools. Versatile properties of general analytical functions associated with the second-order cone are exploited. Based on the so-called self-regular proximity functions, new primal-dual Newton methods for solving SOCO problems are proposed. It will be shown that these new large-update IPMs for SOCO enjoy polynomial ${\cal{O}}({\max\{p,q\} N^{(q+1)/{2q}}\log \frac{N}{\e}})$ iteration bounds analogous to those of their LO and SDO cousins, where N is the number of constraining cones and p,q are constants, the so-called growth degree and barrier degree of the corresponding proximity. Our analysis allows us to choose not only a constant q but also a q as large as logN. In this case, our new algorithm has the best known ${\cal{O}}({N^{1/2}\log{N}\log\frac{N}{\e}})$ iteration bound for large-update IPMs.

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.

As indicated in the recent studies about primal-dual interior-point methods (IPMs) based on kernel functions, a kernel function not only serves to determine the search direction and measure the distance of the current iteration point to the $$\mu $$ -center, but also affects the iteration complexity and the practical computational efficiency of the algorithm. In this paper, we propose a new IPM for semidefinite optimization (SDO) based on a parameterized kernel function which is a generalization of the one presented by Bai et al. (Optim Methods Softw 17(6):985–1008, 2002). By using the good properties of the parameterized kernel function, we deduce that the iteration bound for large-update method is $$O(\sqrt{n}\log {n}\log {\frac{n}{\epsilon }})$$ for $$q=O(n)$$ , which is the best known complexity results for such methods. In our knowledge, this result is the first instance of primal-dual interior point method for SDO which involving the kernel function. Some numerical results have been provided.

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. >

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.

Low-thrust interplanetary spacecraft trajectory optimization poses a uniquely difficult problem to solve because of the inherent nonlinearities of the dynamics and constraints as well as the large size of the search space of possible solutions. Tools currently exist that optimize low-thrust interplanetary trajectories, but these tools are rarely openly available to the public, and when they are available they require multiple interfaces between multiple different packages. The goal of this work is to present a new piece of low-thrust interplanetary spacecraft trajectory optimization software that is open-source and entirely self-contained so that more people can have access to the ability to design interplanetary trajectories. To achieve this goal, a gradient-descent based nonlinear programming method, called the interior point method, was used. The nonlinear programming method was chosen so that results from this work could be compared and contrasted with results from Spacecraft Trajectory Optimization Suite (STOpS), which uses heuristics to iterate towards a solution. Interior point methods are popular because of their ability to handle large amounts of equality and inequality constraints, which is a characteristic that is valuable for low-thrust interplanetary spacecraft trajectories. The software developed, Interior Point Optimizer (IP Optimizer), was then validated against test cases with known solutions to ensure that the software delivered the intended results. Lastly, a constraint satisfaction, a minimum-time, and a maximum-final-mass optimization problem were solved and compared with literature to illustrate the advantages of IP Optimizer and the methods it employs. For the constraint satisfaction problem, IP Optimizer was able to find a solution that exactly satisfied the desired terminal constraints whereas STOpS had an error of 2.29 percent. In this case, IP Optimizer had a reduced runtime of 15 percent compared to STOpS as well. When minimizing time for a spacecraft transfer, IP Optimizer improved upon the solution found by STOpS by 5.3 percent. The speed of convergence for IP Optimizer was almost twice as fast as STOpS for this case. These results show that IP Optimizer is faster than STOpS at converging on a solution and the solution it converges to has a better objective value and more accurately satisfies the terminal constraints than STOpS. Lastly, the maximum-final-mass problem resulted in an objective value that was only 0.5 percent lower than the value found in literature.

In this paper, we present a class of large- and small-update primal-dual interior-point methods forsemidefinite optimization based on a parametric kernel function with a trigonometric barrier term.For both versions of the kernel-based interior-point methods, the worst case iteration bounds are established, namely,$O(n^{\frac{2}{3}}\log\frac{n}{\varepsilon})$ and $O(\sqrt{n}\log\frac{n}{\varepsilon})$, respectively.These results match the ones obtained in the linear optimization case.

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.

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

Real-time dynamic optimization of nonlinear systems: A flatness-based approach