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To formulate a real-world optimization problem, it is sometimes necessary to adopt a set of non-linear terms in the mathematical formulation to capture specific operational characteristics of that decision problem. However, the use of non-linear terms generally increases computational complexity of the optimization model and the computational time required to solve it. This motivates the scientific community to develop efficient transformation and linearization approaches for the optimization models that have non-linear terms. Such transformations and linearizations are expected to decrease the computational complexity of the original non-linear optimization models and, ultimately, facilitate decision making. This study provides a detailed state-of-the-art review focusing on the existing transformation and linearization techniques that have been used for solving optimization models with non-linear terms within the objective functions and/or constraint sets. The existing transformation approaches are analyzed for a wide range of scenarios (multiplication of binary variables, multiplication of binary and continuous variables, multiplication of continuous variables, maximum/minimum operators, absolute value function, floor and ceiling functions, square root function, and multiple breakpoint function). Furthermore, a detailed review of piecewise approximating functions and log-linearization via Taylor series approximation is presented. Along with a review of the existing methods, this study proposes a new technique for linearizing the square root terms by means of transformation. The outcomes of this research are anticipated to reveal some important insights to researchers and practitioners, who are closely working with non-linear optimization models, and assist with effective decision making.

In this work we derive explicit descriptions for the convex envelope of nonlinear functions that are component-wise concave on a subset of the variables and convex on the other variables. These functions account for more than 30 % of all nonlinearities in common benchmark libraries. To overcome the combinatorial difficulties in deriving the convex envelope description given by the component-wise concave part of the functions, we consider an extended formulation of the convex envelope based on the Reformulation–Linearization-Technique introduced by Sherali and Adams (SIAM J Discret Math 3(3):411–430, 1990 ). Computational results are reported showing that the extended formulation strategy is a useful tool in global optimization.

In this work, we introduce and use an innovative approach for adaptive piecewise linear interval approximation of sensor characteristics, which are differentiable functions. The aim is to obtain a discreet type of inverse sensor characteristic, with a predefined maximum approximation error, with minimization of the number of points defining the characteristic, which in turn is related to the possibilities for using microcontrollers with limited energy and memory resources. In this context, the results from the study indicate that to overcome the problems arising from the resource constraints of smart devices, appropriate “lightweight” algorithms are needed that allow efficient connectivity and intelligent management of the measurement processes. The method has two benefits: first, low-cost microcontrollers could be used for hardware implementation of the industrial sensor devices; second, the optimal subdivision of the measurement range reduces the space in the memory of the microcontroller necessary for storage of the parameters of the linearized characteristic. Although the discussed computational examples are aimed at building adaptive approximations for temperature sensors, the algorithm can easily be extended to many other sensor types and can improve the performance of resource-constrained devices. For prescribed maximum approximation error, the inverse sensor characteristic is found directly in the linearized form. Further advantages of the proposed approach are: (i) the maximum error under linearization of the inverse sensor characteristic at all intervals, except in the general case of the last one, is the same; (ii) the approach allows non-uniform distribution of maximum approximation error, i.e., different maximum approximation errors could be assigned to particular intervals; (iii) the approach allows the application to the general type of differentiable sensor characteristics with piecewise concave/convex properties.

This paper investigates the control problem of spacecraft rendezvous with obstacle constraint, considering the external disturbance forces caused by orbit perturbation. Firstly, the translational dynamic model of spacecraft rendezvous is given and then rewritten into a second‐order fully‐actuated system form. Then, by employing the prescribed performance control method, the performance function and error transformation are determined, pre‐defining the prescribed performance bounds. Moreover, the fully‐actuated system approach is used to linearize the original nonlinear system, which simplifies the processes of control law design and ensures model accuracy. After that, to ensure that the spacecraft could avoid the dangerous zone during its manoeuvre, the artificial potential function is introduced, based on which a sliding mode surface is designed. Finally, the prescribed performance control–artificial potential function‐based control law is derived, further adopting the neuro‐adaptive method to deal with external interferences. The stability of the close‐loop control system is analysed through the Lyapunov approach and the effectiveness of the proposed control scheme is verified by carrying out a numerical simulation. This paper investigates the control problem of spacecraft rendezvous with obstacle constraint, considering the external disturbance forces caused by orbit perturbation. The prescribed performance control–artificial potential function‐based control law is derived, adopting the neuroadaptive method to deal with external interferences.

The linearity of active mixers is usually determined by the input transistors, and many works have been proposed to improve it by modified input stages at the cost of a more complex structure or more power consumption. A new linearization method of active mixers is proposed in this paper; the input 1 dB compression point (IP1dB) and output 1 dB compression point (OP1dB) are greatly improved by exploiting the “reverse uplift” phenomenon. Compared with other linearization methods, the proposed one is simpler, more efficient, and sacrifices less conversion gain. Using this method, an ultra-high-linearity double-balanced down-conversion mixer with wide IF bandwidth is designed and fabricated in a 130 nm SiGe BiCMOS process. The proposed mixer includes a Gilbert-cell, a pair of phase-adjusting inductors, and a Marchand-balun-based output network. Under a 1.6 V supply voltage, the measurement results show that the mixer exhibits an excellent IP1dB of +7.2~+10.1 dBm, an average OP1dB of +5.4 dBm, which is the state-of-the-art linearity performance in mixers under a silicon-based process, whether active or passive. Moreover, a wide IF bandwidth of 8 GHz from 3 GHz to 11 GHz was achieved. The circuit consumes 19.8 mW and occupies 0.48 mm2, including all pads. The use of the \"reverse uplift\" allows us to implement high-linearity circuits more efficiently, which is helpful for the design of 5G high-speed communication transceivers.

The reconfiguration problem (DPSR) in electrical distribution systems is a critical area of research, aimed at optimizing the operational efficiency of these networks. Historically, this problem has been approached through a variety of optimization methods. Regarding mathematical models, a key challenge identified in these models is the formulation of equations that ensure the radial operation of the system, along with the nonlinear equations representing Kirchhoff’s laws, the last often necessitating complex relaxations for practical application. This paper introduces an alternative representation of system radiality, which potentially surpasses or matches the existing methods in the literature. Our approach utilizes a more intuitive and compact set of equations, simplifying the representation process. Additionally, we propose a linearization of the current calculation in the power flow model typically used to solve DPSR. This linearization significantly accelerates the process of obtaining feasible solutions and optimal reconfiguration profiles. To validate our approach, we conducted rigorous computational comparisons with the results reported in the existing literature, using a variety of test cases to ensure robustness. Our computational results demonstrate a considerable improvement in computational time. The objective functions used are competitive and, in many instances, outperform the best reported results in the literature. In some cases, our method even identifies superior solutions.

The popularity of smart sensors and the Internet of Things (IoT) is growing in various fields and applications. Both collect and transfer data to networks. However, due to limited resources, deploying IoT in real-world applications can be challenging. Most of the algorithmic solutions proposed so far to address these challenges were based on linear interval approximations and were developed for resource-constrained microcontroller architectures, i.e., they need buffering of the sensor data and either have a runtime dependency on the segment length or require the sensor inverse response to be analytically known in advance. Our present work proposed a new algorithm for the piecewise-linear approximation of differentiable sensor characteristics with varying algebraic curvature, maintaining the low fixed computational complexity as well as reduced memory requirements, as demonstrated in a test concerning the linearization of the inverse sensor characteristic of type K thermocouple. As before, our error-minimization approach solved the two problems of finding the inverse sensor characteristic and its linearization simultaneously while minimizing the number of points needed to support the characteristic.

Additive metal manufacturing (AM), particularly Wire Arc Additive Manufacturing (WAAM), offers a compelling alternative to traditional machining methods. While AM presents advantages such as reduced material waste and lower production costs, challenges remain in effectively controlling the process to prevent defects and optimise material deposition. This article proposes a multivariable control system for WAAM utilising Quantitative Feedback Theory (QFT) to maintain the shape of the heat‐affected zone (HAZ) during transitions in direction flips during layer deposition. By modelling these direction flips as predictable disturbances, the full potential of QFT to integrate feedback and feedforward actions is exploited. The resulting multivariable control laws seek to minimise temperature variation in two critical points around the welding pool by adequately manipulating the power and speed of the heat source. A benchmark system is established to evaluate the effectiveness of the proposed control system. The results demonstrate significant improvement in temperature control, leading to enhanced layer construction quality and reduced need for height corrections or cooling pauses. A multivariable robust control system is proposed for Wire Arc Additive Manufacturing utilizing Quantitative Feedback Theory (QFT) to maintain the shape of the heat‐affected zone (HAZ) in direction flips during layer deposition.

The reformulation-linearization technique (RLT) is a prominent approach to constructing tight linear relaxations of non-convex continuous and mixed-integer optimization problems. The goal of this paper is to extend the applicability and improve the performance of RLT for bilinear product relations. First, we present a method for detecting bilinear product relations implicitly contained in mixed-integer linear programs, which is based on analyzing linear constraints with binary variables, thus enabling the application of bilinear RLT to a new class of problems. Strategies for filtering product relations are discussed and tested. Our second contribution addresses the high computational cost of RLT cut separation, which presents one of the major difficulties in applying RLT efficiently in practice. We propose a new RLT cutting plane separation algorithm which identifies combinations of linear constraints and bound factors that are expected to yield an inequality that is violated by the current relaxation solution. This algorithm is applicable to RLT cuts generated for all types of bilinear terms, including but not limited to the detected implicit products. A detailed computational study based on independent implementations in two solvers evaluates the performance impact of the proposed methods.

We study linear programming relaxations of nonconvex quadratic programs given by the reformulation–linearization technique (RLT), referred to as RLT relaxations. We investigate the relations between the polyhedral properties of the feasible regions of a quadratic program and its RLT relaxation. We establish various connections between recession directions, boundedness, and vertices of the two feasible regions. Using these properties, we present a complete description of the set of instances that admit an exact RLT relaxation. We then give a thorough discussion of how our results can be converted into simple algorithmic procedures to construct instances of quadratic programs with exact, inexact, or unbounded RLT relaxations.