Linear Constraints Research Articles

Identification of labeled Petri nets from finite automata

Automata and Petri nets are two typical models of discrete event systems. The paper studies the problem of converting a finite automaton into a Petri net that satisfies the specified structural features. More specifically, we identify a labeled Petri net from a nondeterminstic finite automaton such that the reachability graph of the identified net is isomorphic to the given automaton, meaning that the Petri net models exactly the same dynamic system as the automaton. The identified net necessarily satisfies the specified structural requirements, such as the numbers of places and transitions, absence of self-loops, the exact structure of a part of the net, and so on. Since label Petri nets and nondeterminstic finite automata are generalizations of Petri nets and deterministic finite automata, respectively, the proposed approach can identify a larger spectrum of Petri nets and effectively handle the nondeterminstic cases in the given automaton. Four kinds of conditions are first extracted from the given automaton, namely deterministic enabling conditions, nondeterministic enabling conditions (i.e., multiple transitions with the same label are enabled simultaneously at a marking), transition disabling conditions, and marking inequality conditions. Then, an integer linear programming problem is formulated by characterizing these four kinds of conditions using integer linear constraints. A labeled Petri net satisfying the structural requirements is obtained by solving the integer linear programming problem. We also present two examples to show the possible applications of the proposed identification approach.

Slater conditions without interior points for programs in Lebesgue spaces with pointwise bounds and finitely many constraints

We consider optimization problems in Lebesgue spaces with pointwise box constraints and finitely many additional linear constraints. We prove that the existence of a Slater point which lies strictly between the pointwise bounds and which satisfies the linear constraints is sufficient for the existence of Lagrange multipliers. Surprisingly, the Slater point is also necessary for the existence of Lagrange multipliers in a certain sense. We also demonstrate how to handle additional finitely many nonlinear constraints.

Chance-Constrained Optimization for a Green Multimodal Routing Problem with Soft Time Window under Twofold Uncertainty

This study investigates a green multimodal routing problem with soft time window. The objective of routing is to minimize the total costs of accomplishing the multimodal transportation of a batch of goods. To improve the feasibility of optimization, this study formulates the routing problem in an uncertain environment where the capacities and carbon emission factors of the travel process and the transfer process in the multimodal network are considered fuzzy. Taking triangular fuzzy numbers to describe the uncertainty, this study proposes a fuzzy nonlinear programming model to deal with the specific routing problem. To make the problem solvable, this study adopts the fuzzy chance-constrained programming approach based on the possibility measure to remove the fuzziness of the proposed model. Furthermore, we use linear inequality constraints to reformulate the nonlinear equality constraints represented by the continuous piecewise linear functions and realize the linearization of the nonlinear programming model to improve the computational efficiency of problem solving. After model processing, we can utilize mathematical programming software to run exact solution algorithms to solve the specific routing problem. A numerical experiment is given to show the feasibility of the proposed model. The sensitivity analysis of the numerical experiment further clarifies how improving the confidence level of the chance constraints to enhance the possibility that the multimodal route planned in advance satisfies the real-time capacity constraint in the actual transportation, i.e., the reliability of the routing, increases both the total costs and carbon emissions of the route. The numerical experiment also finds that charging carbon emissions is not absolutely effective in emission reduction. In this condition, bi-objective analysis indicates the conflicting relationship between lowering transportation activity costs and reducing carbon emissions in routing optimization. The sensitivity of the Pareto solutions concerning the confidence level reveals that reliability, economy, and environmental sustainability are in conflict with each other. Based on the findings of this study, the customer and the multimodal transport operator can organize efficient multimodal transportation, balancing the above objectives using the proposed model.

Simplicial techniques for operator solutions of linear constraint systems

A linear constraint system is specified by linear equations over the group Zd of integers modulo d. Their operator solutions play an important role in the study of quantum contextuality and non-local games. In this paper, we use the theory of simplicial sets to develop a framework for studying operator solutions of linear systems. Our approach refines the well-known group-theoretical approach based on solution groups by identifying these groups as algebraic invariants closely related to the fundamental group of a space. In this respect, our approach also makes a connection to the earlier homotopical approach based on cell complexes. Within our framework, we introduce a new class of linear systems that come from simplicial sets and show that any linear system can be reduced to one of that form. Then we specialize in linear systems that are associated with groups. We provide significant evidence for a conjecture stating that for odd d every linear system admitting a solution in a group admits a solution in Zd.

Decentralized control using selectors for optimal steady-state operation with changing active constraints

We study the optimal steady-state operation of processes where the active constraints change. The aim of this work is to eliminate or reduce the need for a real-time optimization layer, moving the optimization into the control layer by switching between appropriately selected controlled variables (CVs) in a simple way. The challenge is that the best CVs, or more precisely the reduced cost gradients associated with the unconstrained degrees of freedom, change with the active constraints. This work proposes a framework based on decentralized control that operates optimally in all active constraint regions, with region switching mediated by selectors. A key point is that the nullspace associated with the unconstrained cost gradient needs to be selected in accordance with the constraint directions so that selectors can be used. A main benefit is that the number of SISO controllers that need to be designed is only equal to the number of process inputs plus constraints. The main assumptions are that the unconstrained cost gradient is available online and that the number of constraints does not exceed the number of process inputs. The optimality and ease of implementation are illustrated in a simulated toy example with linear constraints and a quadratic cost function. In addition, the proposed framework is successfully applied to the nonlinear Williams–Otto reactor case study.

Impact of Static Urban Traffic Flow-Based Traffic Weighted Multi-Maps Routing Strategies on Pollutant Emissions

Addressing urban traffic congestion is a pressing issue requiring efficient solutions that need to be analyzed regarding travel time and pollutant emissions. The traffic weighted multi-maps (TWM) method has been proposed as an efficient mechanism for congestion mitigation that enables differential traffic routing and path diversity by strategically distributing different network views (maps) to the drivers. Previous works have focused on TWM generation by creating optimal edge weights, but the complexity exponentially increases with the network size and traffic group diversity. This work describes how congestion and emissions can be addressed using TWM maps based on the k-shortest paths for the traffic flows (instead of individuals) that are optimally assigned and distributed to the components of the traffic flow. The map allocation strategies optimal TWM (OTV), optimal TWM per path flow with linear constraints (LCTV), and its variant unconstrained optimal TWM per path flow (UCTV) are described. They use maps generated from the k-shortest paths of the traffic flows (kSP-TWM). The heuristic solution obtained is compared with the theoretical static traffic assignment estimation baseline with different configurations, regarding congestion reduction, total travel time enhancement, and pollutant emissions. Experiments are developed using a synthetic traffic grid network scenario with a mesoscopic simulation. They show that the solution provided is adequate for its proximity to the theoretical equilibrium solutions and can generate minimum emissions patterns. The presented solution opens new possibilities for further congestion and pollutant management studies and seamless integration with existing traffic management frameworks.

A Dual Bounding Framework Through Cost Splitting for Binary Quadratic Optimization

Binary quadratic programming (BQP) is a class of combinatorial optimization problems comprising binary variables, quadratic objective functions, and linear/nonlinear constraints. This paper examines a unified framework to reformulate a BQP problem with linear constraints to a new BQP with an exponential number of variables defined on a graph. This framework relies on the concept of stars in the graph to split the quadratic costs into adjacent and nonadjacent components indicating in-star and out-of-star interactions. We exploit the star-based structure of the new reformulation to develop a decomposition-based column generation algorithm. In our computational experiments, we evaluate the performance of our methodology on different applications with different quadratic structures. The quadratic component of the problem is dealt with in the column generation master problem and its subproblem. Results indicate the superiority of the framework over one of the state-of-the-art solvers, GUROBI, when applied to various benchmark reformulations with adjacent-only or sparse quadratic cost matrices. The framework outperforms GUROBI in terms of both dual bound and computational time in almost all instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: The authors thank the Mitacs Accelerate Program for providing funding for this project. In addition, B. Rostami gratefully acknowledges the funding provided by the Canadian Natural Sciences and Engineering Research Council (NSERC) under a Discovery Grant [Grant RGPIN-2020-05395]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2021.0186 .

Topology optimization for rigid and compliant hybrid mechanisms

This paper proposes a topology optimization method integrating the variable trajectory constraints to design rigid and compliant hybrid mechanisms. The variable trajectory constraints are distributed in the design domain and are uniformly modeled by nonlinear spring model. Each of the variable trajectory constraints has an active or inactive state and different states map various mechanical properties of the nonlinear springs. The nonlinear relation between force and displacement of the spring is formulated. A new group of design variables denoting the states of the springs is introduced into the traditional density-based topology optimization model. The design variables representing the element density of the continuum structure and the states of the variable trajectory constraints are uniformly penalized to avoid intermediate values in order to obtain a physically realizable mechanism. Sensitivity analysis is performed by the adjoint equation method. To demonstrate the versatility and the generality of the proposed method, the typical numerical examples including linear, spline and circular trajectory constraints were implemented. The numerical comparisons between the nonlinear spring model and the actual trajectory constraints verify the accuracy. The proposed method expands the application of topology optimization and can be extensively employed to design the mechanisms integrating both the compliant members and rigid parts.

Melodic Organization and Sequential Ordering of Galant Schemata

This article discusses the organization of galant schemata (Gjerdingen 2007) and its significance across the continuum of music-making from composition to improvisation (Nettl 1974). In Section 1, we argue that galant schemata should be defined more clearly as skeletal core tones, outlines for diminution, and surface formulas after Rabinovitch 2018, 2019a, and 2020a (see also Baragwanath 2020). We compare this organization briefly to additional musical systems discussed by Alaghband-Zadeh (2012), Stoia (2013), Mavromatis (2019), and Carter-Enyi (2021). In Section 2, we describe heuristics for moving from surface to individual soprano core tones, demonstrating how schematic melodic skeletons can be derived (bottom-up) through contrapuntal reduction. In Section 3, we address the sequential ordering of schemata within a first reprise or galant exposition (Burstein 2020) in the 1740s, based on a sample of 27 sections analyzed by hand according to the heuristics. We present a backbone of schemata featured in at least twelve of the pieces, which amounts to a tight script for schema successions. The emergent outline has implications for understanding schemata within form (Byros 2013, 2015; Caplin 2015; Neuwirth 2020; see also Burstein 2020), the linear constraints on skeletons, and schema successions in creative processes (Gjerdingen 2007; Sanguinetti 2012): we argue that it is possible to reconstruct broader linear and formal paths by tracking schema types recurring frequently in the sample. In Section 4, we discuss selected movements from the sixth collection of keyboard pieces for connoisseurs and amateurs by C. P. E. Bach, speculating on their mixture of compositional and improvisatory elements.

Deep Transfer Learning Based Surrogate Modeling for Optimal Investment Decision of Distribution Networks

At present, the distribution network investment is faced with complicated operation strategies and various new investment elements. It's difficult to evaluate the impact of different investment measures on the final benefits due to the complicated physical modeling and potential coupling correlation, which contributes to that the traditional model-driven decision-making method may not work. This paper proposes a novel deep transfer learning based surrogate-assisted method to address the challenge. The key idea is to develop a surrogate model to replace the input-output correlation between the certain investment measure and its benefits, which can be further reformulated into mixed-integer linear constraints and embedded into the classical investment decision-making model for the target distribution network. As the model is trained through a designed two-stage adaptive transfer learning algorithm, which can extract the similar rules from other distribution networks, the dilemma of insufficient effective samples is alleviated, while a neural network constrained algorithm is designed to linear the surrogate model for simplifying the optimization. Numerical cases prove the effectiveness of the surrogate model on correlation mining and show the superiority of the surrogate-assisted method in achieving the precision investment for target distribution network.

Concurrent process and feedrate scheduling with convoluted basis functions and its application to fluid jet polishing

Non-traditional laser and fluid jet processes exhibit time-dependent material removal characteristics. The feedrate profile must be planned carefully along the toolpath for accurate surface profile generation while ensuring that the kinematic limits of machine tools are not violated. Conventional methods iteratively solve a deconvolution/convolution problem on the dwell-time density (reciprocal of the feedrate profile) that is computationally heavy, may leave significant residual processing errors, and even generate infeasible feed profiles with the manufacturing equipment. This paper proposes a novel approach that fully addresses the shortcomings above. Dwell-time density is first expressed as a continuous B-spline profile. The associated dwell basis functions (DBF) are convolved with the process influence function (PIF) to generate new process basis functions (PBF). This approach conveniently allows the posing of the problem as a concurrent linear least-squares problem on the control points shared by the DBFs and PBFs while ensuring the numerical stability of the solution and smoothness of the feed profile. To mitigate excessive acceleration peaks and any ringing effect around the edges of the toolpath, this paper also presents methodologies for stabilizing the scheduled feedrate profile by introducing knot vector adjustments (adaptive knot dropping) and linear edge constraints. The effectiveness of the proposed method is demonstrated and validated through simulation case studies and experimentally in fluid jet processing of precision optics. Results indicate that the proposed technique overcomes the limitations of conventional strategies and allows high-frequency surface components beyond the first zero-power frequency of the process footprint to be tracked while still generating a smooth feed profile within the acceleration limits of a machine tool. This ability stems from the localization characteristics associated with the basis functions. By improving the accuracy of high-frequency components, the proposed method exhibits the potential to fabricate topographies with sharper edges, which has been a challenge for conventional techniques.

Underwater image enhancement using multi-task fusion.

Underwater images are often scattered due to suspended particles in the water, resulting in light scattering and blocking and reduced visibility and contrast. Color shifts and distortions are also caused by the absorption of different wavelengths of light in the water. This series of problems will make the underwater image quality greatly impaired, resulting in some advanced visual work can not be carried out underwater. In order to solve these problems, this paper proposes an underwater image enhancement method based on multi-task fusion, called MTF. Specifically, we first use linear constraints on the input image to achieve color correction based on the gray world assumption. The corrected image is then used to achieve visibility enhancement using an improved type-II fuzzy set-based algorithm, while the image is contrast enhanced using standard normal distribution probability density function and softplus function. However, in order to obtain more qualitative results, we propose multi-task fusion, in which we solve for similarity, then we obtain fusion weights that guarantee the best features of the image as much as possible from the obtained similarity, and finally we fuse the image with the weights to obtain the output image, and we find that multi-task fusion has excellent image enhancement and restoration capabilities, and also produces visually pleasing results. Extensive qualitative and quantitative evaluations show that MTF method achieves optimal results compared to ten state-of-the-art underwater enhancement algorithms on 2 datasets. Moreover, the method can achieve better results in application tests such as target detection and edge detection.

Two-stage robust optimization of unified power quality conditioner (UPQC) siting and sizing in active distribution networks considering uncertainty of loads and renewable generators

This paper proposes a robust optimization siting and sizing strategy for the unified power quality conditioner (UPQC) in the active distribution network (ADN) integrated with renewable energy generators. The optimization objectives including investment cost, line loss, and voltage deviation are integrated into an overall cost for both the planning and operating stages. A novel UPQC equivalent power injection model is proposed based on the auxiliary branch method. A set of linear and second-order conic (SOC) constraints are employed to accurately describe the steady-state operation of UPQC. An improved branch current flow model is developed to incorporate radial distribution networks with UPQCs. The UPQC siting and sizing planning is formulated as a mixed integer second-order conic programming (MISOCP) problem. Accounting for the uncertainty in loads and distributed generators, a two-stage robust optimization model is established to address worst-case operational scenarios. A solving strategy is developed based on the strong duality theory and the column-and-constraint generation (C&CG) algorithm, and then commercial solvers can solve the problem accurately and efficiently. Case studies on an IEEE 33-node distribution network demonstrate this strategy's robustness in dealing with loads and renewable energy uncertainties, making it applicable across different UPQC topologies and operation modes. The proposed method has superior performance over the representative meta-heuristic approaches in terms of global optimality as well as an over 7.5 times faster computation speed. This work has significance in enhancing the quality and efficiency of modern power systems highly integrated with renewable energy.

DMCVS: Decomposed motion compensation‐based video stabilization

AbstractWith the popularity of handheld devices, video stabilization is becoming increasingly important. In previous studies, many methods have been proposed to stabilize shaky videos. However, these methods fail to balance between image content integrity and stability. Some methods sacrifice image content for better stability. Other methods ignore the subtle jitters, which leads to poor stability. This work innovatively proposes a video stabilization method based on decomposed motion compensation. First, a grid‐based motion statistics method is adopted for motion estimation, which obtains more accurate motion vectors according to matched likelihood estimates. Then, the motion compensation is inherently decomposed into two parts: linear motion compensation and auxiliary motion compensation. Linear motion compensation removes complex jitter by constructing linear path constraints to obtain a more stable camera path. Auxiliary motion compensation uses a moving average filter to remove the high‐frequency jitter as a supplement and preserve more image content. The two components are combined with individual weights to derive the final transform matrix and warp the original frames. Experimental results show that our method outperforms the previous methods on NUS and DeepStab datasets qualitatively and quantitatively.

Portfolio selection problem: main knowledge and models (A systematic review)

The challenge in portfolio optimization lies in creating a collection of assets that attains a target expected returnwhile mitigating risk. This problem is often framed as an optimization task, specifically Mean Variance Optimization (MVO). MVO involves formulating an objective function, typically quadratic, that is contingent on the composition of the portfolio, and linear constraints that represents the portfolio's asset allocation restriction. Several improvements have been proposed, such as adding constraints to MVO or using alternative risk measures. As a result, even though MVO model remains the most widely studied type of portfolio optimization, different types of portfolio optimization models, risk/return measurements and constraints have been suggested and used since its invention. In this work, we delve into the various risk and return measures, constraints, and mathematical models commonly used in portfolio optimization.We discuss the key risk measures employed in portfolio optimization, including the Sharpe ratio, beta, maximum drawdown and others, We explore the constraints commonly applied in portfolio optimization. Furthermore, we delve into the mathematical models utilized in portfolio optimization. Then, we emphasize the interplay between risk and return measures, constraints, and mathematical models in portfolio optimization. By providing a comprehensive overview of risk and return measures, constraints, and mathematical models, this work aims to enhance the understanding of portfolio optimization techniques and facilitate informed decisionmaking in the field of investment management. To illustrate different knowledge and models, several experiments were conducted on well-known real data portfolios.

Piecewise non-linear model predictive control with bounded disturbance

This work addresses the problem of robust [Formula: see text] model predictive control for constrained piecewise non-linear systems corrupted by norm-bounded disturbances for the first time. In this approach, the system can have different operating points with different subregions. In each subregion, the system is made up of an affine model disturbed by an additive non-linear term that is locally Lipschitz. The employment of the piecewise non-linear model leads to a non-convex optimization problem, which is far more difficult to solve. It is also much more challenging to develop the [Formula: see text] model predictive control for piecewise non-linear systems. The proposed method introduces a control strategy in the form of a convex optimization problem subject to linear matrix inequality constraints, with the ability to minimize the L2 gain between the disturbance input and the controlled output. The proposed controller guarantees system stability with a prescribed [Formula: see text] disturbance attenuation level under switching between subregions. Simulations on a highly non-linear chemical process are conducted to demonstrate the efficacy of the proposed approach.

An Adaptive Linear Programming Algorithm with Parameter Learning

When dealing with engineering design problems, designers often encounter nonlinear and nonconvex features, multiple objectives, coupled decision making, and various levels of fidelity of sub-systems. To realize the design with limited computational resources, problems with the features above need to be linearized and then solved using solution algorithms for linear programming. The adaptive linear programming (ALP) algorithm is an extension of the Sequential Linear Programming algorithm where a nonlinear compromise decision support problem (cDSP) is iteratively linearized, and the resulting linear programming problem is solved with satisficing solutions returned. The reduced move coefficient (RMC) is used to define how far away from the boundary the next linearization is to be performed, and currently, it is determined based on a heuristic. The choice of RMC significantly affects the efficacy of the linearization process and, hence, the rapidity of finding the solution. In this paper, we propose a rule-based parameter-learning procedure to vary the RMC at each iteration, thereby significantly increasing the speed of determining the ultimate solution. To demonstrate the efficacy of the ALP algorithm with parameter learning (ALPPL), we use an industry-inspired problem, namely, the integrated design of a hot-rolling process chain for the production of a steel rod. Using the proposed ALPPL, we can incorporate domain expertise to identify the most relevant criteria to evaluate the performance of the linearization algorithm, quantify the criteria as evaluation indices, and tune the RMC to return the solutions that fall into the most desired range of each evaluation index. Compared with the old ALP algorithm using the golden section search to update the RMC, the ALPPL improves the algorithm by identifying the RMC values with better linearization performance without adding computational complexity. The insensitive region of the RMC is better explored using the ALPPL—the ALP only explores the insensitive region twice, whereas the ALPPL explores four times throughout the iterations. With ALPPL, we have a more comprehensive definition of linearization performance—given multiple design scenarios, using evaluation indices (EIs) including the statistics of deviations, the numbers of binding (active) constraints and bounds, the numbers of accumulated linear constraints, and the number of iterations. The desired range of evaluation indices (DEI) is also learned during the iterations. The RMC value that brings the most EIs into the DEI is returned as the best RMC, which ensures a balance between the accuracy of the linearization and the robustness of the solutions. For our test problem, the hot-rolling process chain, the ALP returns the best RMC in twelve iterations considering only the deviation as the linearization performance index, whereas the ALPPL returns the best RMC in fourteen iterations considering multiple EIs. The complexity of both the ALP and the ALPPL is O(n2). The parameter-learning steps can be customized to improve the parameter determination of other algorithms.

Distributed continuous-time optimization for convex problems with coupling linear inequality constraints

Distributed continuous-time optimization for convex problems with coupling linear inequality constraints

Comparison of predictors under constrained general linear model and its future observations

. This study deals with some basic inference problems about future observations in a general linear model (GLM) with linear parameter constraints, known as a constrained general linear model (CGLM). Combining the CGLM and its future observations, the author turns the model into a reparameterized form. Using some quadratic matrix optimization methods, the author derives analytical formulas for calculating the best linear unbiased predictors (BLUPs) of all unknown parameter matrices under a CGLM and its future observations. In particular, the author next gives a comprehensive search on the comparison of dispersion matrices of BLUPs of unknown vectors by establishing various equalities and inequalities for dispersion matrices of BLUPs under the model by using elementary block matrix operations and some formulas of rank and inertia of block matrices.

COMPASS: Double-ended saddle point search as a constrained optimization problem.

We present an algorithm to find first order saddle points on the potential energy surface (PES). The algorithm is formulated as a constrained optimization problem that involves two sets of atomic coordinates (images), a time-varying distance constraint and a constraint on the energy difference. Both images start in different valleys of the PES and are pulled toward each other by gradually reducing the distance. The search space is restricted to the pairs of configurations that share the same potential energy. By minimizing the energy while the distance shrinks, a minimum of the constrained search space is tracked. In simple cases, the two images are confined to their respective sides of the barrier until they finally converge near the saddle point. If one image accidentally crosses the barrier, the path is split at suitable locations and the algorithm is repeated recursively. The optimization is implemented as a combination of a quasi-Newton optimization and a linear constraint. The method was tested on a set of Lennard-Jones-38 cluster transitions and a set of 121 molecular reactions using density functional theory calculations. The efficiency in terms of energy and force evaluation is better than with competing methods as long as they do not switch to single-ended methods. The construction of a continuous search path with small steps and the ability to focus on arbitrary subsegments of the path provide an additional value in terms of robustness and flexibility.