Linear Objective Function Research Articles

Orbit-finite linear programming

An infinite set is orbit-finite if, up to permutations of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of linear inequalities. As our principal contribution we provide a decision procedure for checking if such a system has a real solution, and for computing the minimal/maximal value of a linear objective function over the solution set. We also show undecidability of these problems in case when only integer solutions are considered. Therefore orbit-finite linear programming is decidable, while orbit-finite integer linear programming is not.

Finding the optimal probe state for multiparameter quantum metrology using conic programming

The ultimate precision in quantum sensing could be achieved using optimal quantum probe states. However, current quantum sensing protocols do not use probe states optimally. Indeed, the calculation of optimal probe states remains an outstanding challenge. Here, we present an algorithm that efficiently calculates a probe state for correlated and uncorrelated measurement strategies. The algorithm involves a conic program, which minimizes a linear objective function subject to conic constraints on a operator-valued variable. Our algorithm outputs a probe state that is a simple function of the optimal variable. We prove that our algorithm finds the optimal probe state for channel estimation problems, even in the multiparameter setting. For many noiseless quantum sensing problems, we prove the optimality of maximally entangled probe states. We also analyze the performance of 3D-field sensing using various probe states. Our work opens the door for a plethora of applications in quantum metrology.

Investigation of home appliances industry and devices obsolescence considering energy consumption

The objective of this study is to optimize a linear objective function that pertains to the obsolescence of home appliances in a household. The approach introduced is an ILP-based method that is both simple and adaptable. This optimization guarantees that the total energy consumption remains below a predetermined energy threshold, while also adhering to an inequality constraint. The proposed methodology was effectively tested using a household containing four home appliances: a washing machine, a dishwasher, a refrigerator, and an oven. The findings indicate that the refrigerator is the most hazardous and inexpensive home appliance. The current formulation aims to optimize the device's lifespan considering the customer's financial constraints and the expenses associated with repairing and replacing its parts. This study centers on the maintenance of printed circuit boards (PCBs) and considers the effects of degradation-related malfunctions, as assessed through testing. Moreover, we employ an additional ILP approach to assess the vulnerability of refrigerator’s PCB components by determining which ones should be replaced when a customer has limited financial resources.

Data-driven estimation of the algebraic Riccati equation for the discrete-time inverse linear quadratic regulator problem

In this paper, we propose a method for estimating the algebraic Riccati equation (ARE) with respect to an unknown discrete-time system from the system state and input observation. The inverse optimal control (IOC) problem asks, “What objective function is optimized by a given control system?” The inverse linear quadratic regulator (ILQR) problem is an IOC problem that assumes a linear system and quadratic objective function. The ILQR problem can be solved by solving a linear matrix inequality that contains the ARE. However, the system model is required to obtain the ARE, and it is often unknown in fields in which the IOC problem occurs, for example, biological system analysis. Our method directly estimates the ARE from the observation data without identifying the system. This feature enables us to economize the observation data using prior information about the objective function. We provide a data condition that is sufficient for our method to estimate the ARE. We conducted a numerical experiment to demonstrate that our method can estimate the ARE with less data than system identification if the prior information is sufficient.

Pengoptimalan Masalah Nonlinier dalam Meminimumkan Biaya Produksi Menggunakan Separable Programming dan Algoritma Genetika

This research aims to form a nonlinear model of the objective function in the case of minimizing production costs and the number of products that must be produced by Lanting Bumbu An-Nisa. The application of the separable programming method is carried out by transforming the nonlinear objective function and constraints to produce a linear objective function and constraints which are then solved by applying the genetic algorithm method. The application of this method produces a solution that producers must produce 250 packages of onion flavored lanting, 750 packages of cheese flavored lanting, 500 packages of sweet and spicy lanting and 500 packages of corn flavored lanting with a production cost of IDR 12,706,037.29 . The nonlinear model formed in this problem was also solved directly using the genetic algorithm method which resulted in the solution that the total production of onion flavored lanting was 533 packages, 507 packages of cheese flavored lanting, 505 packages of sweet and spicy lanting and 455 packages of corn flavored lanting at a cost of the production that must be spent is IDR 11,213,943.55. The application of these two methods results in a difference in production costs of IDR 1,492,093.74. Based on these results, it shows that solving the nonlinear model directly using a genetic algorithm results in production costs that are 11.74% lower than the costs solved using separable programming.

Constrained State Regulation Problem of Descriptor Fractional-Order Linear Continuous-Time Systems

This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order 0<α<1. The objective is to establish the existence of conditions for a linear feedback control law within state constraints and to propose a method for solving the CSRP of DFOLCS. First, based on the decomposition and separation method and coordinate transformation, the DFOLCS can be transformed into an equivalent fractional-order reduced system; hence, the CSRP of the DFOLCS is equivalent to the CSRP of the reduced system. By means of positive invariant sets theory, Lyapunov stability theory, and some mathematical techniques, necessary and sufficient conditions for the polyhedral positive invariant set of the equivalent reduced system are presented. Models and corresponding algorithms for solving the CSRP of a linear feedback controller are also presented by the obtained conditions. Under the condition that the resulting closed system is positive, the given model of the CSRP in this paper for the DFOLCS is formulated as nonlinear programming with a linear objective function and quadratic mixed constraints. Two numerical examples illustrate the proposed method.

Optimal control in reducing side effects during and after chemotherapy of solid tumors

In this paper, we discuss the effect of optimal chemotherapy doses on solid tumor during treatment and cancer recurrence after the end of treatment. First, an optimal control problem is designed. Then, by changing the control range, the objective function, and adding a control to the previous problem, three new optimal control problems are constructed. We pursue two critical ideas for the success of the drug delivery system to the patient: minimizing the normalized global population density of cancer cells during treatment and at the end of treatment and minimizing side effects during treatment. The optimal control problem is considered with two linear and quadratic objective functions. These models are solved by a well‐established numerical method with high accuracy and a low discretization cost. The numerical results have confirmed that reducing the total dose of the cytotoxic drugs during the treatment period leads to a rapid recurrence of cancer shortly after the end of the treatment period.

Linear optimization on varieties and Chern-Mather classes

The linear optimization degree gives an algebraic measure of complexity of optimizing a linear objective function over an algebraic model. Geometrically, it can be interpreted as the degree of a projection map on the affine conormal variety. Fixing an affine variety, our first result shows that the geometry of this conormal variety, expressed in terms of bidegrees, completely determines the Chern-Mather classes of the given variety. We also show that these bidegrees coincide with the linear optimization degrees of generic affine sections.

Research on the pricing and replenishment decision-making problem of vegetable products based on neural networks and grey wolf algorithm

This article deals with the knapsack problem, where, under real-world conditions, the wholesale prices for the day are not known, and the sales of vegetable products exhibit seasonal patterns. Therefore, this article utilizes an LSTM neural network model to predict the daily wholesale prices. Subsequently, the top 60 projects with the highest profits are selected to create a histogram, and available projects are chosen from these 60.Initially, this article identifies five factors that impact profit and transforms them into positive indicators. Weight allocation is done using the entropy weight method and fuzzy comprehensive evaluation method. The TOPSIS method is employed to obtain composite score indicators for the 60 projects. A linear programming objective function is established, and the results are solved using the Grey Wolf Optimization (GWO) algorithm.

Исследование методов решения задач смешанного целочисленного линейного программирования

The paper discusses mathematical methods and software designed to solve optimization problems with linear objective functions and constraints, subject to additional restrictions on the integrality of variables. The main algorithms for combinatorial optimization are outlined and a comparative analysis of current packages and solvers for solving mixed integer linear programming problems in Python is carried out.

A dual method for polar cuts in disjoint bilinear programming

As one branch of deterministic approaches to disjoint bilinear programming, cutting plane methods are renowned for its ability to systematically reduce the search space by adding cutting planes that are able to cut off regions deemed infeasible or suboptimal. Polar cuts have been widely utilized as a dominating type of cut in terms of deepness. During the establishment of a polar cut, the modified Newton's method is employed to derive the cutting points along the positive or negative extensions of edges emanating from a local solution. Nonetheless, its performance can be further improved along the positive extensions. Drawing inspiration from integer programming, we develop a new approach based on the LP duality theory for this purpose. It re-formulates the original program with a piece-wise linear concave objective function as a single LP. Moreover, we propose a new technique to derive the edges as accommodation to degeneracy. Numerical results show that, by utilizing our newly developed dual method, computing time can be gradually saved as the percentage of generated cutting points along the positive extensions of edges rises.

Scaling Package Queries to a Billion Tuples via Hierarchical Partitioning and Customized Optimization

A package query returns a package---a multiset of tuples---that maximizes or minimizes a linear objective function subject to linear constraints, thereby enabling in-database decision support. Prior work has established the equivalence of package queries to Integer Linear Programs (ILPs) and developed the SketchRefine algorithm for package query processing. While this algorithm was an important first step toward supporting prescriptive analytics scalably inside a relational database, it struggles when the data size grows beyond a few hundred million tuples or when the constraints become very tight. In this paper, we present Progressive Shading, a novel algorithm for processing package queries that can scale efficiently to billions of tuples and gracefully handle tight constraints. Progressive Shading solves a sequence of optimization problems over a hierarchy of relations, each resulting from an ever-finer partitioning of the original tuples into homogeneous groups until the original relation is obtained. This strategy avoids the premature discarding of high-quality tuples that can occur with SketchRefine. Our novel partitioning scheme, Dynamic Low Variance, can handle very large relations with multiple attributes and can dynamically adapt to both concentrated and spread-out sets of attribute values, provably outperforming traditional partitioning schemes such as kd-tree. We further optimize our system by replacing our off-the-shelf optimization software with customized ILP and LP solvers, called Dual Reducer and Parallel Dual Simplex respectively, that are highly accurate and orders of magnitude faster.

When Nash Meets Stackelberg

This article introduces a class of Nash games among Stackelberg players (NASPs), namely, a class of simultaneous noncooperative games where the players solve sequential Stackelberg games. Specifically, each player solves a Stackelberg game where a leader optimizes a (parametrized) linear objective function subject to linear constraints, whereas its followers solve convex quadratic problems subject to the standard optimistic assumption. Although we prove that deciding if a NASP instance admits a Nash equilibrium is generally a [Formula: see text]-hard decision problem, we devise two exact and computationally efficient algorithms to compute and select Nash equilibria or certify that no equilibrium exists. We use NASPs to model the hierarchical interactions of international energy markets where climate change aware regulators oversee the operations of profit-driven energy producers. By combining real-world data with our models, we find that Nash equilibria provide informative, and often counterintuitive, managerial insights for market regulators. This paper was accepted by Chung Piaw Teo, optimization. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.03418 .

OPTIMIZING HARD QOS AND SECURITY WITH DISJOINT PATH ROUTING

Background. The combination of secure routing and hard QoS is a worthwhile topic that involves designing and implementing network protocols and systems that can provide high performance and robust protection for data flow due to shared goals. Secure QoS routing over disjoint paths is a challenging problem that requires balancing the trade-off between network security and bandwidth guarantees.
 Objective. This article investigates a mathematical model that can address secure QoS routing by formulating it as an optimization problem with a linear objective function and linear or bilinear constraints. The objective function aims to minimize the paths compromise probability, while the constraints ensure that the total bandwidth of the paths meets the QoS requirements.
 Methods. We use computer simulation of hard QoS and security with disjoint path routing. Also we use mathematical programming methods in order to describe secure QoS routing.
 Results. The article presents a numerical study of the model using different scenarios and parameters. The results show that the model can effectively provide secure QoS routing over disjoint paths with a high bandwidth guarantee level and a low compromise probability. The work analyses the sensitivity of the solutions to the QoS requirements and reveals that there is usually some margin in the bandwidth provision.
 Conclusions. The proposed model is a promising tool for secure QoS routing over disjoint paths in various network environments.

Minimization of transportation costs using linear programming

This paper intricately explores the utilization of Linear Programming (LP), a distinguished mathematical optimization technique, within the logistics and transportation sector, a domain persistently pursuing methods to bolster efficiency and curtail expenses. LP, characterized by its adeptness in optimizing a linear objective function subject to linear constraints, emerges as a pragmatic solution. This exposition elucidates how LP can be harnessed to ascertain the most economically viable transportation routes and quantities of goods transported from warehouses to consumers. A case study is introduced to spotlight the pragmatic applicability of LP in authentic scenarios, and the potential fiscal savings corporations can realize through its adoption. While recognizing the merits of LP, such as clarity, versatility, and computational efficacy, the paper also sheds light on its limitations, involving its dependency on certain presumptions, a necessity for precise data, and its concentration on single-objective optimization. The discourse concludes by prospecting the future of LP in transportation, exploring its amalgamation with artificial intelligence, machine learning, multi-objective optimization, and green logistics, intending to underscore the significance of LP in transportation and furnish insights for ensuing research and application.

Assessing environmental and operational efficiencies: a multi-objective optimization problem in a two-stage network data envelopment analysis

Abstract Accepted by: Ali Emrouznejad The environmental efficiency of industries plays an important role in economic development of countries. Accordingly, dividing the internal network structure of industries into two sub-processes, including green and operational stages, enables decision-makers to assess both of the efficiencies simultaneously. Such assessment can be implemented using a non-parametric methodology termed data envelopment analysis (DEA). Standard DEA models consider the whole system of decision-making units (DMUs) as a single process (i.e. black-box). The black-box approach ignores modelling of the internal network structure of the assessed DMUs. This issue tackled by network DEA models since it considers the internal network structure of DMUs. In the network DEA, the efficiency evaluation of system stages is essential to identify its overall efficiency, resulting to a multi-objective optimization problem. Therefore, the network DEA is a widely welcomed methodology proposed for solving multi-objective problems. This paper assesses the operational and environmental efficiencies of a network structure system by converting the multi-objective optimization problem into a linear single objective function. In this investigation, a technique of tri-objective function problem is proposed. The proposed technique transforms into a single objective function by keeping one objective function and shifting the other two objective functions into the model’s constraints. The applicability and usefulness of the proposed technique have been tested using a data set of 20 industries. The developed approach provides valuable evaluations to decision-makers to rank DMUs by considering their green and operational efficiency simultaneously.

Fixed-Index-Set Based Approach for Solving a Bilevel Fuzzy Relation Programming With Addition-Min Composition

In recent years, the addition-min fuzzy relation inequalities have been adopted to describe the flow constraint in a P2P network system. Each solution of the inequalities represents a feasible flow control scheme. Motivated by some different managerial objectives, several optimization problems subject to the addition-min inequalities have been recently studied. For example, considering the total efficiency for decreasing the network congestion, the linear objective function was adopted. While considering the fairness among the terminals, the min-max objective function was employed. In this work, combining these two objectives, we establish a corresponding bi-level fuzzy relation programming subject to the addition-min inequalities. For solving the bi-level programming, we first investigate some properties of the first-level programming, using the concept of fixed index set. Based on the properties of the first-level programming, our studied bi-level programming could be equivalently converted into a single-level programming and then solved by the existing linear programming approach. Our resolution approach is carried out by the fixed-index-set based algorithm and illustrated by some numerical examples.

Computing Minimal Boolean Models of Gene Regulatory Networks.

Models of gene regulatory networks (GRNs) capture the dynamics of the regulatory processes that occur within the cell as a means to understanding the variability observed in gene expression between different conditions. Arguably the simplest mathematical construct used for modeling is the Boolean network, which dictates a set of logical rules for transition between states described as Boolean vectors. Due to the complexity of gene regulation and the limitations of experimental technologies, in most cases knowledge about regulatory interactions and Boolean states is partial. In addition, the logical rules themselves are not known a priori. Our goal in this work is to create an algorithm that finds the network that fits the data optimally, and identify the network states that correspond to the noise-free data. We present a novel methodology for integrating experimental data and performing a search for the optimal consistent structure via optimization of a linear objective function under a set of linear constraints. In addition, we extend our methodology into a heuristic that alleviates the computational complexity of the problem for datasets that are generated by single-cell RNA-Sequencing (scRNA-Seq). We demonstrate the effectiveness of these tools using simulated data, and in addition a publicly available scRNA-Seq dataset and the GRN that is associated with it. Our methodology will enable researchers to obtain a better understanding of the dynamics of GRNs and their biological role.

Self-Directed Mobile Robot Navigation Based on Functional Firefly Algorithm (FFA)

This paper proposes an optimized mobile robot navigation strategy using a functional firefly algorithm (FFA) and choice function. This approach has two key advantages: first, the linear objective function performs efficiently with the single degree and finite-order polynomial time operation, and second, the cartesian constraint performs compactly with the chosen degree of freedom on the finite interval. This functional approach optimizes the size of operational parameters in context with key size, operation time, and a finite range of verification. The choice function achieves parameter order (size) reduction. The attraction characteristic of fireflies is represented by the choice function for optimizing the choice between low and high intensities of fireflies. In 2D and 3D environments, the proposed robot navigation performs well in an uncertain environment with static and dynamic obstacles. This efficiency includes the robot’s speed as determined by the choice function’s minimum path lengths. The collision-free path is achieved by the non-void family of non-void sets. The obtained results are optimal in terms of path length and navigational time. The proposed controller is also compared with the other existing controllers, and it is observed that the FFA gives the shortest path in less time for the same environmental condition.

Adaptive warehouse storage location assignment with considerations to order-picking efficiency and worker safety

ABSTRACT Smart warehouses require software-based decision-making tools to manage the receiving, storing, and picking of products. A major challenge in achieving efficient operations is deciding where to store products associated with incoming orders. The storage location assignment problem (SLAP) is more complex in large-size warehouses due to several functional objectives and numerous possible shelving solutions. This paper introduces an artificial intelligence algorithm that seeks to find an acceptable solution to SLAP with presented linear and nonlinear objective functions. The near-optimal technique exploits basin-hopping and simulated-annealing algorithms to find a solution when considering four functional objectives including worker safety, which has not been optimized using similar approaches. The algorithm is experimentally evaluated, and results demonstrate that reasonablely achieved solutions are comparable to those obtained by well-known existing solvers. Furthermore, the problem could be solved with non-linear objectives which is beyond the commercial solvers’ like SCIP capability.