Piecewise Linearization Techniques Research Articles

Machine learning-based methods for piecewise digital predistortion in mmW 5G NR systems

Piecewise linearization techniques require dividing the signal into multiple pieces each linearized individually. Machine learning (ML) is one of the useful tools to perform the automatic division of these pieces. Complexity reduction in the classification of piecewise digital predistortion is possible through carefully constructing features from both the signal statistics and the power amplifier (PA) characteristics. Our paper introduces two low-complex classical ML-based methods that facilitate the classification of baseband input data into distinct segments. These methods effectively linearize PA behavior by employing tailored Volterra models corresponding to each segment. Moreover, we perform an in-depth analysis of the proposed schemes to further optimize their classification and regression complexities. The two proposed low-complexity approaches are validated by laboratory experiments and show up to 4 dB error vector magnitude (EVM) improvement over the conventional approach for a class A PA at 28 GHz. Similarly, the EVM improvement is up to 2 dB over the vector-switched general memory polynomial scheme. With only one indirect learning architecture iteration, the two proposed schemes obey the 5G new radio standard up to 6.5 dB and 7 dB output backoff, respectively.

Incorporating catastrophe insurance in power distribution systems investment and planning for resilience enhancement

Severe damage and high economic losses caused by frequent extreme disasters are becoming unbearable for distribution system operators (DSOs) under climate change. In order to cope with extreme disasters, this paper proposes an investment portfolio and planning framework of multi-resource for resilience enhancement. In the investment portfolio and planning framework, insurance is incorporated to mitigate high economic losses as a risk sharing and resilience enhancement supplementary strategy, and co-optimized with security investment from the perspective of DSOs. Firstly, the insurance investment is modelled based on Gordon-Loeb model according to insurance industry principles and formulated as constraints by piecewise linearization techniques. Secondly, the investment portfolio and planning framework proposed in this paper is modeled as a scenario-based two-stage stochastic mixed integer linear optimization program (MILP). In the first stage, DSO makes investment and planning decisions including energy storage (ESS) allocation, line hardening strategy and insurance investment. Both operation strategies of normal operation scenarios and extreme disaster scenarios under climate change are optimized in the second stage. Finally, progressive hedging (PH) algorithm is applied to solve the two-stage stochastic MILP model. The proposed model and algorithm are tested on the modified IEEE 33-bus test system. The results verify the validity of investment portfolio and planning framework, and the results show that portfolio investment considering insurance can realize resilience enhancement in both load shedding and economic losses.

Comparison of optimization frameworks for the design of a multi-energy microgrid

The scope of the paper is to investigate different strategies for the design of a multi-energy system considered as a systemic optimization problem. The objective is to determine the best sizes of the energy assets such as electrochemical and thermal storages, cogeneration units, solar generators and chillers. In these cases, the techno-economic optimization is a tradeoff between the operating costs and the capital expenditures in the form of integrated management and design of the system. The paper addresses the challenges of these optimization problems in two steps. The former implements generic piecewise linearization techniques based on non-linear models. That approach allows a significant reduction of the computational time for the management loop of the assets (i.e. optimal power dispatch). The latter takes into consideration the integration of that management loop in different architectures for optimal system planning. The main contribution of the paper toward filling the gap in the literature is to investigate a wide range of optimization frameworks - with bi-level optimizations (using both deterministic and evolutionary methods), Monte-Carlo simulations as well as a performant ‘all-in-one’ approach in which both sizes and controls are variables of a single mathematical problem formulation. Finally, a thorough results analysis highlights that the best solution tends to be the same whether the objective to optimize is the traditional net present value at the end of the system lifespan or the total yearly cost of ownership.

Energy Flow Optimization for Integrated Power–Gas Generation and Transmission Systems

This paper first presents a comprehensive model of the gas system with detailed formulations on pipelines, short pipes, resistors, valves, compressors, and compressor stations. Furthermore, an optimal energy flow model is proposed for integrated power-gas generation and transmission systems. Specifically, on the generation side, gas-fired units couple the two energy systems as the power generation and gas sink; on the transmission side, gas compressor stations link the two energy systems as the power demand and gas transportation. However, gas flow equations are nonlinear and gas flow directions also need to be optimized. Logical programming and tailored piecewise linearization techniques are performed, leading to a mixed-integer linear program (MILP). Only a logarithmic number of binary variables are introduced to represent the nonlinear quadratic function, and thus, the MILP model can be solved very efficiently. Numerical results on four power-gas test systems demonstrate the effectiveness of the proposed approach.

Optimal integrated sizing and planning of hubs with midsize/large CHP units considering reliability of supply

Use of multi-carrier energy systems and the energy hub concept has recently been a widespread trend worldwide. However, most of the related researches specialize in CHP systems with constant electricity/heat ratios and linear operating characteristics. In this paper, integrated energy hub planning and sizing is developed for the energy systems with mid-scale and large-scale CHP units, by taking their wide operating range into consideration. The proposed formulation is aimed at taking the best use of the beneficial degrees of freedom associated with these units for decreasing total costs and increasing reliability. High-accuracy piecewise linearization techniques with approximation errors of about 1% are introduced for the nonlinear two-dimensional CHP input-output function, making it possible to successfully integrate the CHP sizing. Efficient operation of CHP and the hub at contingencies is extracted via a new formulation, which is developed to be incorporated to the planning and sizing problem. Optimal operation, planning, sizing and contingency operation of hub components are integrated and formulated as a single comprehensive MILP problem. Results on a case study with midsize CHPs reveal a 33% reduction in total costs, and it is demonstrated that the proposed formulation ceases the need for additional components/capacities for increasing reliability of supply.

A Relative Entropy Approach to Stochastic Dominance Analysis

This study formulates portfolio analysis in terms of Stochastic Dominance, Relative Entropy and Empirical Likelihood. We define a portfolio inefficiency measure based on the divergence between given probabilities and the nearest probabilities that rationalize a given portfolio for some admissible utility function. When applied to a sample of time-series observations in a blockwise fashion, the inefficiency measure becomes a Likelihood Ratio statistic for inequality moment conditions. The limiting distribution of the test statistic is bounded by a chi-squared distribution under general sampling schemes, allowing for conservative large-sample testing. We develop a tight numerical approximation for the test statistic based on a two-stage optimization procedure and piecewise linearization techniques. A Monte Carlo simulation study of the Empirical Likelihood Ratio test shows superior small-sample properties compared with various Generalized Method of Moments tests. An application analyzes the efficiency of a passive stock market index in data sets from the empirical asset pricing literature.

Portfolio Analysis Using Stochastic Dominance, Relative Entropy, and Empirical Likelihood

This study formulates portfolio analysis in terms of stochastic dominance, relative entropy, and empirical likelihood. We define a portfolio inefficiency measure based on the divergence between given probabilities and the nearest probabilities that rationalize a given portfolio for some admissible utility function. When applied to a sample of time-series observations in a blockwise fashion, the inefficiency measure becomes a likelihood ratio statistic for testing inequality moment conditions. The limiting distribution of the test statistic is bounded by a chi-squared distribution under general sampling schemes, allowing for conservative large-sample testing. We develop a tight numerical approximation for the test statistic based on a two-stage optimization procedure and piecewise linearization techniques. A Monte Carlo simulation study of the empirical likelihood ratio test shows superior small-sample properties compared with various generalized method of moments tests. An application analyzes the efficiency of a passive stock market index in data sets from the empirical asset pricing literature. This paper was accepted by Manel Baucells, decision analysis.

On solving three-dimensional open-dimension rectangular packing problems

ABSTRACTIn this article, a recently proposed three-dimensional open-dimension rectangular packing problem is considered, in which the objective is to find a minimal volume rectangular container that packs a set of rectangular boxes. The literature has tackled small-sized instances of this problem by means of optimization solvers, position-free mixed-integer programming (MIP) formulations and piecewise linearization approaches. In this study, the problem is alternatively addressed by means of grid-based position MIP formulations, whereas still considering optimization solvers and the same piecewise linearization techniques. A comparison of the computational performance of both models is then presented, when tested with benchmark problem instances and with new instances, and it is shown that the grid-based position MIP formulation can be competitive, depending on the characteristics of the instances. The grid-based position MIP formulation is also embedded with real-world practical constraints, such as cargo stability, and results are additionally presented.

Optimal Design of a Speed Reducer

The mathematical model for optimal design of a speed reducer is a generalized geometric programming problem that is non-convex and not easy be globally solved. This paper applies a deterministic approach including convexification strategies and piecewise linearization techniques to globally solve speed reducer design problems. A practical speed reducer design problem is solved to demonstrate that this study obtains a better solution than other methods.

Design Optimization of a Speed Reducer Using Deterministic Techniques

The optimal design problem of minimizing the total weight of a speed reducer under constraints is a generalized geometric programming problem. Since the metaheuristic approaches cannot guarantee to find the global optimum of a generalized geometric programming problem, this paper applies an efficient deterministic approach to globally solve speed reducer design problems. The original problem is converted by variable transformations and piecewise linearization techniques. The reformulated problem is a convex mixed-integer nonlinear programming problem solvable to reach an approximate global solution within an acceptable error. Experiment results from solving a practical speed reducer design problem indicate that this study obtains a better solution comparing with the other existing methods.

An Efficient Multiperiod MINLP Model for Optimal Planning of Offshore Oil and Gas Field Infrastructure

In this paper, we present an efficient strategic/tactical planning model for offshore oilfield development problem that is fairly generic and can be extended to include other complexities. The proposed multiperiod nonconvex MINLP (mixed integer nonlinear programming) model for multifield site includes three components (oil, water, and gas) explicitly in the formulation using higher order polynomials, avoiding bilinear and other nonlinear terms. With the objective of maximizing total NPV for long-term planning horizon, the model involves decisions related to FPSO (floating production, storage, and offloading) installation and expansions, field–FPSO connections, well drilling, and production rates in each time period. The model can be solved effectively with DICOPT for realistic instances and gives good quality solutions. Furthermore, it can be reformulated into an MILP after piecewise linearization and exact linearization techniques that can be solved globally in an efficient way. Solutions of realistic in...

Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems

Many global optimization approaches for solving signomial geometric programming problems are based on transformation techniques and piecewise linear approximations of the inverse transformations. Since using numerous break points in the linearization process leads to a significant increase in the computational burden for solving the reformulated problem, this study integrates the range reduction techniques in a global optimization algorithm for signomial geometric programming to improve computational efficiency. In the proposed algorithm, the non-convex geometric programming problem is first converted into a convex mixed-integer nonlinear programming problem by convexification and piecewise linearization techniques. Then, an optimization-based approach is used to reduce the range of each variable. Tightening variable bounds iteratively allows the proposed method to reach an approximate solution within an acceptable error by using fewer break points in the linearization process, therefore decreasing the required CPU time. Several numerical experiments are presented to demonstrate the advantages of the proposed method in terms of both computational efficiency and solution quality.

Global Optimization of Gas Lifting Operations: A Comparative Study of Piecewise Linear Formulations

Continuous gas lifting is the process of increasing oil well production by injecting compressed natural gas, called “lift gas”, into the production tubing of an oil well [ Presented at the SPE Gas Technology Symposium, Calgary, Alberta, Canada, 1996]. This paper considers the problem of optimizing the distribution of a limited supply of lift gas to wells in an oil field using piecewise linearization techniques. Four modeling approaches, proposed by Nemhauser and Woolsey [Integer and Combinatorial Optimization; J. Wiley: New York, 1988], Foudas [Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications; Oxford University Press: New York, 1995], Sherali [Oper. Res. Lett. 2001, 28, 155.], and Keha et al. [Oper. Res. Lett. 2004, 32, 44.], are presented and the gas lifting problem is solved using each method. Each of the four frameworks is sufficient to solve the problem to global optimality and the method presented by Keha et al. has the best computational performance. The gas lifting problem is...

Finding all solutions of systems of nonlinear equations with free variables

Systems of nonlinear equations often represent mathematical models in engineering design. This study proposes a novel method for finding all solutions of systems of nonlinear equations with free variables. The original problem is first transformed into a global optimization problem whose multiple global minima with a zero objective value correspond to all solutions of the original problem. Then, by using variable substitution on free variables and applying convexification strategies and piecewise linearization techniques on nonconvex functions, the transformed optimization problem is reformulated as a convex mixed-integer program solvable to reach an approximately global optimum. An algorithm is developed to find all solutions of the reformulated problem. Numerical examples in real applications are presented to demonstrate the usefulness of the proposed method in engineering design.

Global optimization of signomial mixed-integer nonlinear programming problems with free variables

Mixed-integer nonlinear programming (MINLP) problems involving general constraints and objective functions with continuous and integer variables occur frequently in engineering design, chemical process industry and management. Although many optimization approaches have been developed for MINLP problems, these methods can only handle signomial terms with positive variables or find a local solution. Therefore, this study proposes a novel method for solving a signomial MINLP problem with free variables to obtain a global optimal solution. The signomial MINLP problem is first transformed into another one containing only positive variables. Then the transformed problem is reformulated as a convex mixed-integer program by the convexification strategies and piecewise linearization techniques. A global optimum of the signomial MINLP problem can finally be found within the tolerable error. Numerical examples are also presented to demonstrate the effectiveness of the proposed method.

A seasonal demand inventory model with variable lead time and resource constraints

In this paper, two new methods are proposed for solving a seasonal demand problem with variable lead-time and resource constraints. Despite its significance, no study has been done on such problem to obtain the best policy. First, in order to solve the variable lead time, a linear programming relaxation using piecewise linearization techniques is derived. Then, a mixed integer program with linearization techniques is constructed for the seasonal demand problem. Finally, some illustrative examples are included to demonstrate the applicability of the proposed models.

A global optimization method for packing problems

The objective of packing problems is to determine an optimal way of placing a given set of three-dimensional (3D) rectangular cartons within a minimum volume 3D rectangular container. Current packing optimization methods either use too many extra 0–1 variables or find it difficult to obtain a globally optimal solution. This study proposes an efficient method for finding the global optimum of packing problems. First the traditional packing optimization problem is converted into an equivalent program containing many fewer 0–1 variables than contained in current methods. Then the global optimum of the converted program is found by utilizing piecewise linearization techniques. The numerical examples demonstrate that the proposed method is capable of finding the global optimum of a packing problem.

On the mixed integer signomial programming problems

This paper proposes an approximate method to solve the mixed integer signomial programming problem, for which the objective function and the constraints may contain product terms with exponents and decision variables, which could be continuous or integral. A linear programming relaxation is derived for the problem based on piecewise linearization techniques, which first convert a signomial term into the sum of absolute terms; these absolute terms are then linearized by linearization strategies. In addition, a novel approach is included for solving integer and undefined problems in the logarithmic piecewise technique, which leads to more usefulness of the proposed method. The proposed method could reach a solution as close as possible to the global optimum.

A linearization approach for inventory models with variable lead time

This paper proposes an approximate global optimization approach for solving inventory models with variable lead time to find a solution close to a global optimum. A linear programming relaxation based on piecewise linearization techniques is derived for the problem. First, it converts non-linear terms into the sum of absolute terms, which are then linearized by goal-programming techniques and linearization approaches. Finally, several typical examples are used to show the superior flexibility of the proposed approach that allows the user to add his own constraints.

On the posynomial fractional programming problems

The posynomial fractional programming (PFP) problem arises from the summation minimization of several quotient terms, which are composed of posynomial terms appearing in the objective function subject to given posynomial constraints. This paper proposes an approximate approach to solving a PFP problem. A linear programming relaxation is derived for the problem based on piecewise linearization techniques, which first convert a posynomial term into the sum of absolute terms; these absolute terms are then linearized by some linearization techniques. The proposed approach could reach a solution as close as possible to a global optimum.