A generalized self-regular Kernel function for large-scale nonlinear optimization problems

Economic dispatching control (EDC) is one of the most important functions of a central load dispatching center of power utilities. The conventional EDC algorithm that employs the equal incremental fuel cost method minimizes the fuel costs subject to the system energy balance. It is, however, unable to treat power unit dynamics and generator constraints that extend over plural time slices. In this paper, we propose a new economic dispatching control algorithm that takes unit dynamics into consideration to lower fuel costs and is able to treat various kinds of constraints using the interior point optimization method.

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.

The development of robust and efficient algorithms for nonlinear optimization is an important matter; because they make possible the solution of models with increasingly sizeand complexity. In this context, the Interior Point Optimizer (IPOPT) has been proven to be a competitive algorithm, because of its ability to handle large scale problems. However, there are some specific situations in which convergence can not be guaranteed. For instancethe violation of a regularity condition given by the Linear Independence Constraint Qualification (LICQ). This property is given in terms of the consistency of local linearizationsof the set of constraints, i.e. the gradients. If the set of gradients of equality and active bound inequality are linearly independent, then the LICQ holds. It turns out that if thisproperty is not satisfied, IPOPT would attempt to regularize the system by perturbing it, with the hope to overcome regions of local inconsistency. However, if the LICQ does nothold globally or if the system has severe ill-conditioned, the regularization strategies will fail. Thus, we present the `1-exact penalty-barrier strategy to deal with this lack of regularity. This works by penalizing the infeasibility with a parameter at the objective; and then by reformulating the problem to obtain a higher dimensional problem, the LICQ property holds at all points for which the new variables are nonzero. Thus achieving higher level of robustness as opposed to the regular IPOPT. Moreover, an unified framework for nonlinear model-based state-estimation and control with parametric sensitivity is presented. Typically, control strategy of a processes is tied to the ability to accommodate online computations; in this sense nonlinear optimization strategies like Moving Horizon Estimation (MHE) and Model Predictive Control (NMPC) are overlooked. Nevertheless, it has been proposed to use the parametric nature of these optimization problems to create a scheme in which the bulk of computations is partitioned into a background phase with all the most computationally expensive steps, and an online phase, in which the sensitivity information is used to make inexpensive calculations for the state estimate or controller input. These ideas have been put together into a Python framework, that streamlines the creation of a state-estimation and control strategy with the additional benefit of having parametric sensitivity embedded into it. This framework works by using the Pyomo libraries to create optimization objects and automatically discretize the models given by the user. We demonstrate the effectiveness of the frameworkin two problems of vastly different complexity. A benchmark distillation case study and a Bubbling Fluidized Bed Reactor (BFB) for the capture of CO2. The results show thatsensitivity is effective for online implementations of the controller and it was seamlessly enabled by our framework.

In this paper,a combined cooling heating and power (CCHP) system model considering energy supply pipe network is proposed. Compared with the traditional CCHP system framework, more types of energy supply equipment and their possible connections are considered during the stage of energy supply station modeling. During operation simulation, energy supply regulation is applied according to daily load profile. Heat loss and pressure loss of the pipe network are also calculated in the process. A Particle Swarm Optimization (PSO) - Iteration (IM) - Interior point (LP) optimization algorithm is applied to the combined planning of energy station and energy supply network of CCHP Micro-system. By this method, the influence of energy supply pipe network model on CCHP Micro-system design and planning, as well as economic efficiency of the whole project are studied.

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.

Recent research shows that non-convex AC OPF problem can be recast as a convex Semidefinite (SD) problem or Second Order Cone Programming (SOCP) problem. This paper presents a stochastic SOCP OPF (SSOCP-OPF) model for power systems connected to the wind farms. This is performed by reformulating the original non-convex OPF problem using more practical parameters of the power system. Finally, we obtain a convex optimization problem through some well-known approximations and an exact relaxation incorporating the stochastic nature of wind power. One of the advantage of convex SOCP problems, which are a general form of linear problems, is that they can be efficiently solved through Interior Point Methods (IPMs). The proposed OPF model takes advantage of both DC-OPF models (solution efficiency) and full AC-OPF models (solution accuracy). As an application of SSOCP-OPF model, we study the impact of wind power uncertainty on the transmission loss in the power systems. To evaluate the proposed stochastic model modified IEEE 30-bus test system is used. The optimization problem is coded in GAMS platform and solved using its embedded interior point optimizer MOSEK.

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.

The four target criteria of lead time, scheduling deviation, work-in-progress (WIP), and utilization determine the objectives of production logistics. As a kind of typical queuing systems, optimization of performance measures is intractable to do in manufacturing systems. Fortunately, owing to its exellent characteristics, the convex optimization method can be very helpful. Based on the queuing theory, a nonlinear load optimal allocation model is proposed in this paper. A novel transformation of the optimization variables is also devised and the constraints are properly combined so as to make this model a convex one. The interior-point method for convex optimization is presented here as a computationally efficient tool. Finally, this model is assessed on a real case, from which such conclusions are drawn that the convex optimization can ensure the full utilization of machines and the least amount of WIP(work-in-progress); other performance measures of manufacturing systems may also be optimized in the similar way.

Dynamical model development and parameter identification for solid-state anaerobic digestion of shellfish products: Application to Mytilus edulis

In this paper, a real-time optimizing controller is developed to steer a differentially flat nonlinear control system to closed-loop trajectories that optimize a cost functional of interest. An interior point optimization method with penalty function is used to formulate a real-time optimization scheme. The problem is posed as a real-time optimal trajectory generaction problem in which the optimal trajectories are computed using an adaptive extremum-seeking approach. A fed-batch bioreactor simulation example is used to demonstrate the application of the technique to a finite time dynamic optimization problems that arise in batch process control.

This paper presents an optimization framework to design the components and the controller of a Continuously Variable Transmission (CVT) in an integrated manner. Specifically, we aim at reducing the mass of the transmission and the leakage losses that occur in the system. To do so, we first formulate the joint plant and control design problem including the corresponding objectives and constraints. Thereafter, we propose a proportional integral structure for the design of the CVT ratio control. The combined plant and control design problem is formulated as a nonlinear multi-objective optimization problem, and is simultaneously solved using an interior point optimization method. We evaluate the obtained design on the Worldwide Harmonized Light Vehicles Test Cycle (WLTC) as well as on more aggressive driving scenarios, and demonstrate that the optimized CVT design is always capable of realizing the required driving performance. Additionally, we study the impact of the plant design parameters on the control performance by analyzing the coupling strength between the subproblems. Thereby, the pulley radius is found to have the strongest influence in the resulting leakage losses that occur at the variator level. Finally, leveraging the presented design framework, we show that up to 13% and 18% reduction in the CVT variator mass and on leakage losses, respectively, can be achieved without compromising the desired ratio trajectory over a representative dynamic driving cycle.

This paper presents the design analysis of a suborbital launch vehicle by performing a multidisciplinary optimization analysis. The design problem consists of establishing the optimized launch scenario using the interior point optimization method. The minimization of the objective function (the total mass of the launch vehicle) is validated by a multi-criteria approach. It is shown that establishing the performance of the launch vehicle should meet the criteria of security and control during the launch mission. The security aspect is represented by adherence to the dynamic pressure for structural matters of the launch vehicle and the acceleration, represented by the G-force, which should remain tolerable. Control is denoted by keeping the vehicle’s velocity within the range required for a suborbital flight mission. The approach followed is appealingly constructive for conserving the multidisciplinary design optimization (MDO) formulation for a post-performance analysis.

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.

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.