Linear Objective 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.

Objective quantification of sound sensory attributes in side-by-side vehicles using multiple linear regression models

The evaluation of sound quality is a pivotal area of research within audio and acoustics. The sound quality evaluation methods commonly used include both objective and subjective, the latter being time-consuming and costly as they rely on listening tests. This research work aims to investigate the use of predictive sound quality models as a way to objectively assess the Desire-to-buy of side-by-side vehicles, in a more efficient, faster, and less costly way than conventional methods. Multiple linear regression algorithms were used to validate the objective models derived from objective physical metrics and perceptual psycho-physical metrics. The sensory profile objective models reported in this paper were constructed using parsimonious linear Lasso and Elastic-net algorithms. Our results show that linear objective models effectively account for each of the perceptual attributes of the sensory profiles and the Desire-to-buy, while only requiring a few physical and psychophysical metrics.

Investigation of Markovian Process in Server Break-down

In day to day life goods produced at a number of places are consumed at different locations and then the problem faced by producer or dealer is to transfer goods from numerous places of origin to various places of consumption such that cost of transportation, time required, damages to the product etc. are optimum. This is called classical transportation problem. In this study we have studied transportation problem solve by Markov Decision Process. Introduction to the operations research is made along with over view of optimization techniques of single objective transportation problem. For literature review in the subject of multi objective fractional as well as linear multi objective transportation problems. In basic concepts of Markov Decision Process are given for further usage in the thesis work. We have stated algorithms for solving different types of multi objective programming problems. In entire work we have used basically linear membership functions, hyperbolic membership function and exponential membership function. These membership functions are Markov Decision Process. In case of linear membership function weighted arithmetic mean, weighted quadratic mean, geometric mean is used to find compromise solution. Almost everywhere we have found that the compromise solutions are fairly close to optimal solution when the problem is solved as single objective function. We have solution procedure for multi objective linear transportation is given using Markov Decision Process. Introduction to fractional programming is made and solutions of multi objective linear fractional programming problems are discussed. We have solutions of multi objective linear fractional transportation problem. In this topic fractional objective functions are approximated as linear functions using Taylor’s theorem and then applied method to solve the multi objective linear transportation problem discussed earlier. We focus on solution to fractional transportation problem with Markov Process coefficients. To multi objective fractional transportation problem with goal programming approach. We deal with general integer programming problem along with fractional transportation problems. The main result of this thesis is a characterization of increasing Marko processes satisfying certain conditions. At the end bibliography and research papers published by authors are given. Keywords: Markov Process, Transportation, Queen analyses Server system.

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.

Sparse graphical designs via linear programming

Graphical designs are a framework for sampling and numerical integration of functions on graphs. In this note, we introduce a method to address the trade-off between graphical design sparsity and accuracy. We show how to obtain sparse graphical designs via linear programming and design objective functions that aim to maximize their accuracy. We showcase our approach using yellow taxicab data from New York City.

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.

A MaxSAT approach for solving a new Dynamic Discretization Discovery model for train rescheduling problems

Train scheduling is a critical activity in rail traffic management, both off-line (timetabling) and on-line (dispatching). Time-Indexed formulations for scheduling problems are stronger than other classical formulations, like Big-M. Unfortunately, their size grows usually very large with the size of the scheduling instance, making even the linear relaxation hard to solve. Moreover, the approximation introduced by time discretization can lead to solutions which cannot be realized in practice. Dynamic Discretization Discovery (DDD), recently introduced by Boland et al. (2017) for the continuous-time service network design problem, is a technique to keep at bay the growth of Time-Indexed formulations and their response times and, at the same time, ensures the necessary modelling precision. By exploiting the DDD paradigm, we develop a novel approach to train dispatching and, more in general, to job-shop scheduling. The algorithm implemented represents the first application of a Maximum SATisfiability problem approach to the field. In our comparisons on real-life instances of train dispatching, our restricted Time-Indexed formulation solves faster on piece-wise constant objective functions, while the Big-M approach maintains the lead on linear continuous objectives.

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.

Multi-battle contests over complementary battlefields

AbstractThis paper studies contests with complementary prizes where each agent simultaneously distributes a fixed budget over multiple battlefields. Each battlefield has a single prize which is divided among the competitors in proportion to an arbitrary power function of their investment levels. A unique pure strategy Nash equilibrium is shown to exist under arbitrarily sensitive battlefield success functions if objective functions exhibit constant subunitary elasticity of substitution between prize shares. In contrast, Blotto contests with linear objectives have only mixed strategy Nash equilibria if battlefield success functions are sufficiently sensitive to investment levels. Sufficient complementarity between prize shares allows pure strategy Nash equilibria to exist under arbitrarily sensitive battlefield success functions.

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.

Adjustability in robust linear optimization

AbstractWe investigate the concept of adjustability—the difference in objective values between two types of dynamic robust optimization formulations: one where (static) decisions are made before uncertainty realization, and one where uncertainty is resolved before (adjustable) decisions. This difference reflects the value of information and decision timing in optimization under uncertainty, and is related to several other concepts such as the optimality of decision rules in robust optimization. We develop a theoretical framework to quantify adjustability based on the input data of a robust optimization problem with a linear objective, linear constraints, and fixed recourse. We make very few additional assumptions. In particular, we do not assume constraint-wise separability or parameter nonnegativity that are commonly imposed in the literature for the study of adjustability. This allows us to study important but previously under-investigated problems, such as formulations with equality constraints and problems with both upper and lower bound constraints. Based on the discovery of an interesting connection between the reformulations of the static and fully adjustable problems, our analysis gives a necessary and sufficient condition—in the form of a theorem-of-the-alternatives—for adjustability to be zero when the uncertainty set is polyhedral. Based on this sharp characterization, we provide two efficient mixed-integer optimization formulations to verify zero adjustability. Then, we develop a constructive approach to quantify adjustability when the uncertainty set is general, which results in an efficient and tight poly-time algorithm to bound adjustability. We demonstrate the efficiency and tightness via both theoretical and numerical analyses.

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.

New Approach to Solving Fuzzy Multiobjective Linear Fractional Optimization Problems

In this paper, an iterative approach based on the use of fuzzy parametric functions is proposed to find the best preferred optimal solution to a fuzzy multiobjective linear fractional optimization problem. From this approach, the decision-maker imposes tolerance values or termination conditions for each parametric objective function. Indeed, the fuzzy parametric values are computed iteratively, and each fuzzy fractional objective is transformed into a fuzzy non-fractional parametric function using these values of parameters. The core value of fuzzy numbers is used to transform the fuzzy multiobjective non-fractional problem into a deterministic multiobjective non-fractional problem, and the ε-constraint approach is employed to obtain a linear single objective optimization problem. Finally, by setting the value of parameter ε, the Dangtzig simplex method is used to obtain an optimal solution. Therefore, the number of solutions is equal to the number of used values, and the optimal solution is chosen according to the preference of the decision-maker. We have provided a didactic example to highlight the step of our approach and its numerical performances.

Generalization of the method based on parameterization developed for solving integer programming problems

This study presents the generalization of methods based on parameterization proposed for solving integer programming problems with linear objectives and linear constraints. Initially, the method based on parameterization, proposed for solving 2, 3, and 4-variable integer linear programming problems, is generalized to n-variable integer linear programming problems. The proposed method relies on the parameterization obtained from Diophantine equations. Using this parameterization, the original problem is reformulated as another integer linear programming problem that can be solved more efficiently using simple mathematical programming. The method can be used regardless of the number of constraints of the problem and also provides all alternative solutions to the decision maker. It is demonstrated with some examples that this method provides an efficient algorithm.

Comparative Statics With Linear Objectives: Normality, Complementarity, and Ranking Multi‐Prior Beliefs

Comparative Statics With Linear Objectives: Normality, Complementarity, and Ranking Multi‐Prior Beliefs

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.