Piecewise Linear Approximation Research Articles

Global optimization of mixed integer signomial fractional programing problems

The main issue in the present article is to investigate how to solve a mixed integer fractional signomial geometric programing problem (MIFSGP). In the first step to achieving this idea, we must convert a fractional signomial programing problem into a non-fractional problem via a simple conversion technique. Then, a convex relaxation with a new modified piecewise linear approximation with integer break points as a pre-solve method is used to reach an integer global optimum solution. A few numerical examples are included to illustrate the advantages of the proposed method

Stability tests for neutral‐type delay systems: A delay Lyapunov matrix and piecewise linear approximation of the functional argument approach

AbstractNew necessary and sufficient stability tests for neutral‐type linear time delay systems are presented. They are based on functionals with prescribed derivative depending on the delay Lyapunov matrix and recent stability/instability theorems. A piecewise linear approximation of the initial function argument in the functional is employed. As a result, the functional is approximated via a quadratic form whose matrix provides stability criteria. This matrix involves the integrals based on the delay Lyapunov matrix which are handled via an efficient computational procedure. The main feature of these tests is that they are verified in a finite number of operations, which is achieved through rigorous bounding of the approximation error on a specific class of functions. Some examples which validate the obtained results are presented and discussed.

A more accurate piecewise linear approximation method for quadratic cost curves of thermal generators and its application in unit commitment

A more accurate piecewise linear approximation method for quadratic cost curves of thermal generators and its application in unit commitment

Estimating fatigue life of carbon/epoxy composites: A rapid method coupling thermo‐mechanical analysis and residual strength

AbstractThe investigation of fatigue behavior typically involves time‐consuming tests, leading some researchers to explore methodologies based on self‐heating tests to reduce the process. For composite materials, the conventional approach involves piecewise linear approximations of the self‐heating curve and the stress transition between the first and second regimes is arbitrarily associated with a fatigue lifetime equal to 106 cycles. This paper proposes a novel methodology to address these simplifications. First, a non‐linear viscoelastic model is used to describe the self‐heating curve. Based on the mechanisms established in the literature, a link is proposed between the dissipation and the fatigue limit. The load leading to a significant contribution of non‐linear mechanisms is associated with an infinite life time. It allows the identification of an – curve and prediction of fatigue behavior through a minimal number of tests. The comparison with fatigue results is satisfactory.

NADOL: Neuromorphic Architecture for Spike-Driven Online Learning by Dendrites.

Biologically plausible learning with neuronal dendrites is a promising perspective to improve the spike-driven learning capability by introducing dendritic processing as an additional hyperparameter. Neuromorphic computing is an effective and essential solution towards spike-based machine intelligence and neural learning systems. However, on-line learning capability for neuromorphic models is still an open challenge. In this study a novel neuromorphic architecture with dendritic on-line learning (NADOL) is presented, which is a novel efficient methodology for brain-inspired intelligence on embedded hardware. With the feature of distributed processing using spiking neural network, NADOL can cut down the power consumption and enhance the learning efficiency and convergence speed. A detailed analysis for NADOL is presented, which demonstrates the effects of different conditions on learning capabilities, including neuron number in hidden layer, dendritic segregation parameters, feedback connection, and connection sparseness with various levels of amplification. Piecewise linear approximation approach is used to cut down the computational resource cost. The experimental results demonstrate a remarkable learning capability that surpasses other solutions, with NADOL exhibiting superior performance over the GPU platform in dendritic learning. This study's applicability extends across diverse domains, including the Internet of Things, robotic control, and brain-machine interfaces. Moreover, it signifies a pivotal step in bridging the gap between artificial intelligence and neuroscience through the introduction of an innovative neuromorphic paradigm.

A 0–1 linear relaxation and stochastic approximation method for real-time pricing in a smart grid with non-convex constraints

In this article, real-time pricing (RTP) for a smart grid with complicated non-convex and/or non-smooth constraints is explored. First, piecewise linear functions are adopted to approximate any kinds of utility function and RTP is formulated into bilevel non-convex programming. Then, a 0–1 relaxation method is used to linearize sparse constraints and piecewise linear utility functions, to obtain mixed integer linear programming. Finally, a distributed simultaneous perturbation stochastic approximation (DSPSA) method is designed to solve the bilevel non-convex programming. The RTP strategy is generalizable because sparse constraints and non-differentiable and/or non-concave utility functions are widespread in practice, and the piecewise linear functions approximation method is applicable to all kinds of utility function, not just to non-smooth and/or non-concave utility functions. In addition, the 0–1 linear relaxation and stochastic approximation method make the bilevel non-convex programming easily solvable by DSPSA, which converges rapidly while maintaining the high accuracy of solutions.

Piecewise linear approximation for MILP leveraging piecewise convexity to improve performance

To realize adaptive operation planning with MILP unit commitment, piecewise-linear approximations of the functions that describe the operating behavior of devices in the energy system have to be computed. We present an algorithm to compute a piecewise-linear approximation of a multi-variate non-linear function. The algorithm splits the domain into two regions and approximates each region with a set of hyperplanes that can be translated to a convex set of constraints in MILP. The main advantage of this “piecewise-convex approximation” (PwCA) compared to more general piecewise-linear approximation with simplices is that the MILP representation of PwCA requires only one auxiliary binary variable. For this reason, PwCA yields significantly faster solving times in large MILP problems where the MILP representation of certain functions has to be replicated many times, such as in unit commitment. To quantify the impact on solving time, we compare the performance using PwCA with the performance of simplex approximation with logarithmic formulation and show that PwCA outperforms the latter by a big margin. For this reason, we conclude that PwCA will be a useful tool to set up and solve large MILP problems such as arise in unit commitment and similar engineering optimization problems.

A linear programming formulation for eco-driving over road slopes

This research proposes a linear program (LP) formulation for energy-minimizing vehicle velocity optimization in hilly terrain and demonstrates comparable performance to its traditional formulation as a dynamic program (DP). The LP captures the kinetic energy dynamics and road grade preview exactly and employs a novel piecewise linear approximation to travel time. The LP runs faster than its DP counterpart and can run in real-time and in closed-loop control applications. To demonstrate its effectiveness, the LP velocity planner is integrated with a receding horizon motion planner and simulated in a Class 8 tractor–trailer application over a real road slope profile, with and without traffic. In these closed-loop simulations, energy savings are observed relative to an explicit cruise control law when maintaining constant speed requires excessive braking.

Two efficient logarithmic formulations to solve nonconvex economic dispatch

To solve economic dispatch (ED) with nonconvex nonlinear generation cost functions, this paper adopts a global optimization technique where piecewise linear approximation (PLA) functions replace the nonlinear functions. However, a problem with this technique is that the PLA model requires incorporating many discrete variables into the optimization model, which may render the model computationally expensive. This paper presents two efficient representation techniques of the PLAs, which scale logarithmically, including binary zig-zag (ZZB) and general integer zig-zag (ZZI). We replace the nonconvex cost functions in the ED with their PLAs and obtain an alternative ED formulation where two logarithmic techniques model the PLAs. Then, one can efficiently solve the alternative ED model with the standard branch-and-cut/bound-based algorithms to the desired optimality gap. To verify the techniques’ efficacy, we consider test systems having 13, 38, 40, 80, and 160 units. The numerical evaluations show that the logarithmic techniques presented can yield high-quality optimal solutions in less than 2 seconds in the considered cases, illustrating the techniques’ efficiency.

Efficient continuous piecewise linear regression for linearising univariate non-linear functions

Due to their flexibility and ability to incorporate non-linear relationships, Mixed-Integer Non-Linear Programming (MINLP) approaches for optimization are commonly presented as a solution tool for real-world problems. Within this context, piecewise linear (PWL) approximations of non-linear continuous functions are useful, as opposed to non-linear machine learning-based approaches, since they enable the application of Mixed-Integer Linear Programming techniques in the MINLP framework, as well as retaining important features of the approximated non-linear functions, such as convexity. In this work, we extend upon fast algorithmic approaches for modeling discrete data using PWL regression by tuning them to allow the modeling of continuous functions. We show that if the input function is convex, then the convexity of the resulting PWL function is guaranteed. An analysis of the runtime of the presented algorithm shows which function characteristics affect the efficiency of the model, and which classes of functions can be modeled very quickly. Experimental results show that the presented approach is significantly faster than five existing approaches for modeling non-linear functions from the literature, at least 11 times faster on the tested functions, and up to a maximum speedup of more than 328,000. The presented approach also solves six benchmark problems for the first time.

Simultaneous topology design and optimization of PDE constrained processes based on mixed integer formulations

Simultaneous topological design and optimization of complex processes that are described by partial differential equations is a challenging but promising research area. Widely adopted nested and sequential approaches are mostly applicable based on heuristic solutions, hindering the theoretical improvement potential due to decentralized decision-making in subsequent stages with a significant number of trial-and-error procedures. This study introduces a mixed integer formulation addressing the governing equations and case-dependent topological constraints at each discretization point, enabling solutions through rigorous solvers under process-related constraints and objectives. Nonlinear expressions in the formulations are further tailored using piecewise linear approximations, still representing the major nonlinear trends through a mixed-integer linear nature to favor global optimality and benefit from computational advancements, when needed. Heat and Stokes flow problems are used as case studies to demonstrate the applicability of the methodology.

Economic Model Predictive Control in Buildings Based on Piecewise Linear Approximation of Predicted Mean Vote Index

Energy shortage is a challenge for many countries, and building energy consumption accounts for a considerable proportion of global energy consumption. The main work of this paper is to optimize the energy consumption of heating, ventilating, and air conditioning (HVAC) systems in buildings based on economic model predictive control (EMPC). The cost in EMPC design includes energy consumption and predicted mean vote (PMV), which is an index that evaluates the thermal comfort of indoor occupants. In order to model the nonlinearity of the PMV index, we propose a lattice piecewise linear (PWL) approximation, which has high approximation precision and facilitates the resulting optimization problem, which is basically a piecewise quadratic programming problem. For the piecewise quadratic programming, we propose a descent algorithm that converges quickly and scales well with the length of the prediction horizon in the EMPC problem. The experimental results demonstrate that the proposed method saves 19.78% of the electricity cost compared to the conventional control strategy and significantly increases indoor comfort. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> — The motivation of this article is to provide a control strategy to reduce building energy consumption and ensure indoor thermal comfort. In most of the existing methods for air conditioning temperature control, the occupants’ comfort hasn’t been considered. In this paper, thermal comfort is described by the PMV index, which is basically nonlinear. In order to model the thermal comfort more accurately, in this paper, the PMV index is approximated piecewise linearly in order to meet the requirements of accuracy and computational efficiency. The resulting optimization problem is not hard to solve, and we provide an efficient algorithm for solving this optimization problem. Preliminary simulation experiments demonstrate that this approach is practical, i.e., it achieves energy reduction and ensures thermal comfort. Our strategy, however, has not yet been deployed in real buildings. In future research, we will propose similar techniques for large-scale systems in order to solve energy optimization problems containing multiple thermal zones and realize the proposed technique in real buildings.

Substantiating the directions of health tourism development in Czechia

&lt;abstract&gt; &lt;p&gt;The purpose of the article is to substantiate the directions of health tourism development in Czechia. The purpose of the study is the subjects of health tourism: well-known rehabilitation centers in Czechia, medical institutions, health resorts. The methodological basis was formed by the following methods: abstract-logical; statistical; expert evaluation; cluster analysis; piecewise linear approximation. Among the most popular regions of Czechia for health tourism, the main places are Prague, Moravian-Silesia, Olomouc, Plzeň, South Moravian and South Bohemian. The increase in the number of patients among residents of the country and foreigners who use health tourism services is studied. The five leaders among the countries that used the services are identified. The main problems and advantages of health tourism in Czechia are identified. The Czech tourism development strategy for 2021–2030 is analyzed and directions for its implementation are proposed. &lt;bold&gt;Originality / value&lt;/bold&gt;: the originality of the study lies in forecasting the directions of health tourism development in Czechia using the method of piecewise linear approximation in the following areas: adults with health insurance; adults at their own expense, children and teenagers, which will increase the volume of income for health tourism institutions in Czechia. According to the forecasting results, the demand for health services will increase by 3% annually. The results of this study can be useful for substantiating the directions of further health tourism development in the country, which will ensure the competitiveness and investment attractiveness of the country.&lt;/p&gt; &lt;/abstract&gt;

Using infeasible path cuts to solve Electric Vehicle Routing Problems with realistic charging functions exactly within a branch-and-cut framework

The paper investigates the Electric Vehicle Routing Problem with a non-linear concave and strictly monotonic increasing charging function. In the literature, the non-linear charging function is typically approximated by a piecewise linear charging function which does not overestimate the real charging function in any point. As the piecewise linear charging function underestimates the real state-of-charge in some points, such an approximation excludes feasible solutions from the solution space. To overcome this drawback we introduce a new method to determine a piecewise linear charging function overestimating the real charging function in a way that the area between both functions is minimized as well as an adaptation of a known linearization to include the piecewise linear charging function in a branch-and-cut approach. Thereby, we include infeasible solutions in the solution space. To declare them infeasible again we check every integer solution obtained in the branch-and-cut procedure and add an infeasible path cut if the solution is infeasible for the real charging function such that the procedure terminates with an optimal solution for the real charging function. Our approach is evaluated in a computational study in which instances with up to 100 customers were solved to optimality. Moreover, we evaluate the trade-off between a more complex model formulation due to more binary variables if the number of supporting points for the piecewise linear approximation is increased and the higher approximation error if fewer supporting points are used.

An algorithm for stochastic convex-concave fractional programs with applications to production efficiency and equitable resource allocation

We propose an algorithm to solve convex and concave fractional programs and their stochastic counterparts in a common framework. Our approach is based on a novel reformulation that involves differences of square terms in the constraints, and subsequent employment of piecewise-linear approximations of the concave terms. Using the branch-and-bound (B&B) framework, our algorithm adaptively refines the piecewise-linear approximations and iteratively solves convex approximation problems. The convergence analysis provides a bound on the optimality gap as a function of approximation errors. Based on this bound, we prove that the proposed B&B algorithm terminates in a finite number of iterations and the worst-case bound to obtain an ϵ-optimal solution reciprocally depends on the square root of ϵ. Numerical experiments on Cobb–Douglas production efficiency and equitable resource allocation problems support that the algorithm efficiently finds a highly accurate solution while significantly outperforming the benchmark algorithms for all the small size problem instances solved. A modified branching strategy that takes the advantage of non-linearity in convex functions further improves the performance. Results are also discussed when solving a dual reformulation and using a cutting surface algorithm to solve distributionally robust counterpart of the Cobb–Douglas example models.

Efficient computational strategies for a mathematical programming model for multi-echelon inventory optimization based on the guaranteed-service approach

This paper presents a Multi-Echelon Inventory Optimization (MEIO) framework, based on the Guaranteed-Service Model (GSM), to allocate safety stocks across a supply chain with several locations and products, minimizing costs while meeting service level objectives. Extending previous work by Achkar et al. (2023), this paper enhances the Mixed-Integer Quadratically Constrained Program (MIQCP) with a highly efficient solution approach. The model introduces a piecewise linear approximation, significantly improving computational efficiency and the accuracy of the approximation for the fill rate function. It also introduces a different and more efficient approach to account for stochastic lead times using a discrete function. Moreover, an extension of the approach to account for non-normally distributed demands is proposed. The model is applied to several instances of a real-world case study from a pharmaceutical company, with more than 7300 product-location combinations, showing that optimal solutions can be obtained within few seconds of computational time.

Neural network informed day-ahead scheduling of pumped hydro energy storage

This paper presents a neural network-constrained optimization model for the optimal scheduling of pumped hydro energy storage. Neural networks are trained offline to capture the complex head-dependent performance curves in both pump and turbine modes using actual operation data. The trained models are then embedded into the optimization framework that yields the optimal and physics-compliant day-ahead scheduling in energy and reserve markets for the pumped hydro energy storage. To identify the trade-off between modeling accuracy and computation burden, different neural network architectures are investigated, along with the impact of neural network sparsity, i.e., weights pruning to reduce dimensionality. The proposed approach is then compared with state-of-the-art solutions, such as piecewise linear approximations. To that end, a detailed simulator of the pumped hydro energy storage, mimicking its minute-wise behavior, is developed to accurately assess the feasibility and economic performance of the resulting schedules. Results demonstrate the ability of neural networks to better guide the optimization model, thus leading to higher profits while keeping acceptable solving times, especially when weight pruning is leveraged. In particular, we show that accurately capturing the non-linear characteristics of pumped hydro energy storage is critical to offer reliable reserve commitments to power systems.

Piecewise reconstruction of membership function approximation errors for Takagi–Sugeno fuzzy control

We propose a conservative relaxed control method to stabilize Takagi–Sugeno (TS) fuzzy systems. First, a conventional piecewise linear function approximation method is adopted to consider the membership functions’ information in the closed-loop system’s stability analysis. Subsequently, a model reconstruction method is proposed to reformulate linear approximation error terms in the TS fuzzy form, which can effectively introduce more information into stability conditions and achieve better robust control effects. Then, based on a fuzzy Lyapunov function approach, control design conditions are derived in terms of linear matrix inequalities, with less conservatism for related results in the literature. Finally, via a benchmark experiment, it is verified that the proposed method can show better robustness with fewer decision variables in robust convex conditions.

Piecewise linear approximation with minimum number of linear segments and minimum error: A fast approach to tighten and warm start the hierarchical mixed integer formulation

In several areas of economics and engineering, it is often necessary to fit discrete data points or approximate non-linear functions with continuous functions. Piecewise linear (PWL) functions are a convenient way to achieve this. PWL functions can be modeled in mathematical problems using only linear and integer variables. Moreover, there is a computational benefit in using PWL functions that have the least possible number of segments. This work proposes a novel hierarchical mixed integer linear programming (MILP) formulation that identifies a continuous PWL approximation with minimum number of linear segments for a given target maximum error. The proposed MILP formulation also identifies the solution with the least maximum error among the solutions with minimum number of segments. Then, this work proposes a fast iterative algorithm that identifies non necessarily continuous PWL approximations by solving O(S log N) linear programming (LP) problems, where N is the number of data points and S is the minimum number of segments in the non necessarily continuous case. This work demonstrates that tight bounds for the MILP problem can be derived from these approximations. Next, a fast algorithm is introduced to transform a non necessarily continuous PWL approximation into a continuous one. Finally, the tight bounds and the continuous PWL approximations are used to tighten and warm start the MILP problem. The tightened formulation is shown in experimental results to be more efficient, especially for large data sets, with a solution time that is up to two orders of magnitude less than the existing literature.

Rolling horizon policies for multi-stage stochastic assemble-to-order problems

Assemble-to-order approaches deal with randomness in demand for end items by producing components under uncertainty, but assembling them only after demand is observed. Such planning problems can be tackled by stochastic programming, but true multistage models are computationally challenging and only a few studies apply them to production planning. Solutions based on two-stage models are often short-sighted and unable to effectively deal with non-stationary demand. A further complication may be the scarcity of available data, especially in the case of correlated and seasonal demand. In this paper, we compare different scenario tree structures. In particular, we enrich a two-stage formulation by introducing a piecewise linear approximation of the value of the terminal inventory, to mitigate the two-stage myopic behaviour. We compare the out-of-sample performance of the resulting models by rolling horizon simulations, within a data-driven setting, characterised by seasonality, bimodality, and correlations in the distribution of end item demand. Computational experiments suggest the potential benefit of adding a terminal value function and illustrate interesting patterns arising from demand correlations and the level of available capacity. The proposed approach can provide support to typical MRP/ERP systems, when a two-level approach is pursued, based on master production and final assembly scheduling.