Linear Optimization Problem Research Articles

Dynamic Resource Allocation: The Geometry and Robustness of Constant Regret

We study a family of dynamic resource allocation problems, wherein requests of different types arrive over time and are accepted or rejected. Each request type is characterized by its reward, arrival probability, and resource consumption. An upper bound for the collected reward is given by a linear optimization problem with a random right-hand side. This type of problem, known as packing linear program (LP), is ubiquitous in resource allocation. We provide a detailed characterization of the parametric structure of this packing LP. Relying on this geometric understanding, we revisit and expand on BudgetRatio algorithms that achieve constant regret by resolving this same packing LP in each period and accepting requests scored as sufficiently valuable. We illustrate the benefits of the geometric view in proving that (i) BudgetRatio achieves constant regret relative to the offline (full information) upper bound in the presence of inventory that is (slowly) restocked, and (ii) within explicitly identifiable bounds, the algorithm’s regret is robust to misspecification of the model parameters. This gives bounds for the bandits version of the problem in which the parameters have to be learned. (iii) The algorithm has an equivalent formulation as a generalized bid-price algorithm in which the bid prices can be adaptively and efficiently computed. Our analysis focuses on the evolution of the remaining inventory—in turn of the LP that drives BudgetRatio—as a stochastic process. We prove that it is attracted to sticky regions of the state space in which the online algorithm takes actions consistent with the optimal basis of the offline upper bound, a basis that is revealed only in hindsight at the horizon’s end. Funding: This work was supported by the U.S. Department of Defense [Grant W911NF-20-C-0008].

Optimal EV Charging and PV Siting in Prosumers towards Loss Reduction and Voltage Profile Improvement in Distribution Networks

In this paper, the problem of simultaneous charging of Electrical Vehicles (EVs) in distribution networks (DNs) is examined in order to depict congestion issues, increased power losses, and voltage constraint violations. To this end, this paper proposes an optimal EV charging schedule in order to allocate the charging of EVs in non-overlapping time slots, aiming to avoid overloading conditions that could stress the DN operation. The problem is structured as a linear optimization problem in GAMS, and the linear Distflow is utilized for the power flow analysis required. The proposed approach is compared to the one where EV charging is not optimally scheduled and each EV is expected to start charging upon its arrival at the residential charging spot. Moreover, the analysis is extended to examine the optimal siting of small-sized residential Photovoltaic (PV) systems in order to provide further relief to the DN. A mixed-integer quadratic optimization model was formed to integrate the PV siting into the optimization problem as an additional optimization variable and is compared to a heuristic-based approach for determining the sites for PV installation. The proposed methodology has been applied in a typical low-voltage (LV) DN as a case study, including real power demand data for the residences and technical characteristics for the EVs. The results indicate that both the DN power losses and the voltage profile are further improved in regard to the heuristic-based approach, and the simultaneously scheduled penetration of EVs and PVs could yield up to a 66.3% power loss reduction.

Linear State Optimal Control Problem with a Stochastic Switching Time

In this paper, we analyse an optimal control problem over a finite horizon with a stochastic switching time, assuming that the two optimal control problems present in its two stages have a particularly simple form called linear state. It is well known that linear state optimal control problems can be solved easily using the HJB equation approach and assuming that the value function is linear in the state. Unfortunately, this simplicity of solution does not extend to the problem with stochastic switching time. We prove that a necessary and sufficient condition for the problem to maintain a linear state structure is to assume that the hazard rate of the switching time depends only on the temporal variable. Finally, assuming that the hazard rate is constant, we completely characterise the solution of the obtained linear state optimal control problem.

Optimal switching of vaccination for an infectious disease model

We study a switched SIHR (Susceptible Infected Hospitalized Recovered) model for infectious diseases. The aim is to find the most effective switching signal to manage the disease's effects as effectively as maintaining a continuous vaccination program. A nonlinear, non-convex optimal control problem of the switched system is converted into a linear and convex optimal control problem by applying the theory of moments and semi-definite programming. Finally, some numerical simulations are performed to compare continuous vaccination programs and optimal switching of vaccination control. For more information see https://ejde.math.txstate.edu/Volumes/2024/59/abstr.html

Partially observable optimal control using exponential cost criterion

We consider a partially observable optimal control model in which the system and observation noises are correlated, and the cost criterion is in the form of an exponential of an integral. A new minimum principle criterion is derived and applied to compute the explicit solution of linear exponential Gaussian optimisation problem with correlated noises.

Invariant set estimation for piecewise affine dynamical systems using piecewise affine barrier function

This paper introduces an algorithm for estimating the invariant set of closed-loop controlled dynamical systems identified using single-hidden layer Rectified linear units (ReLU) neural networks or piecewise affine (PWA) functions, particularly addressing the challenge of providing safety guarantees for single-hidden layer ReLU networks commonly used in safety–critical applications. The invariant set of PWA dynamical system is estimated using single-hidden layer ReLU networks or its equivalent PWA function. This method entails formulating the barrier function as a PWA function and converting the search process into a linear optimization problem using vertices. We incorporate a domain refinement strategy to increase flexibility in case the optimization does not find a valid barrier function. Moreover, the objective of the optimization is to find a less conservative invariant set based on the current partition. Our experimental results demonstrate the effectiveness and efficiency of our approach, demonstrating its potential for ensuring the safety of PWA dynamical systems.

Fractional Strong Metric Dimension of Convex Polytopes and its applications

The fractional versions of various metric related parameters have recently gained importance due to their applications in the fields of sensor networking, robot navigation and linear optimization problems. Convex polytopes are collection of those polytopes of Euclidean space which are their convex subsets. They have key importance in the field of network designing due to their stable and resilient structure which aids optimal data transfer. The identification and removal of components (nodes) of a communication network causing abruption in its flow is of key importance for optimal data transmission. These components are referred as strong resolving neighbourhood (SRNs) in graph theory and assigning least weight to these components aids the computation of fractional strong metric dimension (FSMD). In this paper, we compute FSMD for certain convex polytopes which include $\mathbb{P}_{n}$, $\mathbb{P}_{n}^{1}$ and $\mathbb{P}_{n}^{2}$. In this regard, it is shown that for $n \geq 3$, FSMD of $\mathbb{P}_{n}$ and $\mathbb{P}_{n}^{2}$ is $n$ and $\frac{3n}{2}$, respectively. Also, FSMD of $\mathbb{P}_{n}^{1}$ is $n$ when $n$ is odd and $\frac{3n}{2}$ when $n$ is even. Finally, an application of FSMD in the context of internet connection networks is furnished.

An efficient algorithm for solving multilevel multi-objective linear fractional optimization problem with neutrosophic parameters

This work demonstrates the modeling of the multilevel-multiobjective linear fractional optimization (ML-MOLFO) problem in a neutrosophic decision environment. The primary driving force behind this work is to determine how to handle a ML-MOLFO problem while taking into account a real-life decision-making scenario. There is a hierarchy of decision-makers and many decision-making processes, such as agree, not-sure, and disagree, in a real-life decision-making setting. By taking into account such genuine situations, all resource, coefficient of objective, and constraint functions are represented as single-valued trapezoidal neutrosophic numbers. As far as we can tell, there is no established technique for resolving the ML-MOLFO problem with neutrosophic parameters in the literature review. Therefore, by outlining goal programming methods in the intuitionistic fuzzy environment, we create our new intuitionistic fuzzy Chebyshev goal programming (IFCGP) method to address the introduced problem. In the proposed method, first we used a modified ranking function to transform the problem into an analogous crisp ML-MOLFO problem, and then it was converted into a multilevel multiobjective linear optimization problem through linearized techniques. The linearization procedure in the proposed method, unlike the existing one, does not require derivatives, additional variables, or inequality constraints, which makes it simpler. To illustrate the adequacy and performance of the proposed new method, we considered numerical tests and application problems. We also compared the performance of the resulting optimal solution with existing approaches using the distance metric, which indicates our method is non-dominated.

Geographic Information System for Shortest Route Search for Clinics in Pamekasan Regency Using the Djikstra Method

Earthquake natural disasters can cause significant infrastructure damage, hindering community access and mobility. In such a situation, determining the shortest path for evacuation becomes very important. This research uses linear programming by utilising Excel Solver to determine the shortest evacuation route for earthquake natural disasters. Excel Solver is a feature in Microsoft Excel that serves as an analytical tool to solve linear optimisation problems. Excel solver can be used to find solutions that maximise or minimise the objective function, with respect to predetermined constraints. This research utilises relevant earthquake natural disaster simulation data as constraints in the optimisation model. The results show that the model can generate shortest paths that fulfil various constraints and criteria. The model can also be used to predict the time required to reach the destination. This research makes an important contribution to the development of tools and methods for shortest path optimisation in natural disaster scenarios. The optimisation model developed in this research can be used to assist decision makers in improving the efficiency and effectiveness of evacuation and disaster relief operations.

The GroupMax Neural Network Approximation of Convex Functions.

We present a new neural network to approximate convex functions. This network has the particularity to approximate the function with cuts which is, for example, a necessary feature to approximate Bellman values when solving linear stochastic optimization problems. The network can be easily adapted to partial convexity. We give an universal approximation theorem in the full convex case and give many numerical results proving its efficiency. The network is competitive with the most efficient convexity-preserving neural networks and can be used to approximate functions in high dimensions.

On the relationship between the value function and the efficient frontier of a mixed integer linear optimization problem

On the relationship between the value function and the efficient frontier of a mixed integer linear optimization problem

Optimisation of Evacuation Route Determination in an Earthquake Natural Disaster Scenario Using an Excel Solver

Earthquake natural disasters can cause significant infrastructure damage, hindering community access and mobility. In such a situation, determining the shortest path for evacuation becomes very important. This research uses linear programming by utilising Excel Solver to determine the shortest evacuation route for earthquake natural disasters. Excel Solver is a feature in Microsoft Excel that serves as an analytical tool to solve linear optimisation problems. Excel solver can be used to find solutions that maximise or minimise the objective function, with respect to predetermined constraints. This research utilises relevant earthquake natural disaster simulation data as constraints in the optimisation model. The results show that the model can generate shortest paths that fulfil various constraints and criteria. The model can also be used to predict the time required to reach the destination. This research makes an important contribution to the development of tools and methods for shortest path optimisation in natural disaster scenarios. The optimisation model developed in this research can be used to assist decision makers in improving the efficiency and effectiveness of evacuation and disaster relief operations.

Solution sets of linear fractional optimization problems involving integral functions

We consider a linear fractional optimization problem (FP) involving integral function defined on C^n [0, 1], and then characterize solution sets for the problem (FP) in terms of sequential Lagrange multipliers of a known solution of (FP). Moreover, we give an example illustrating our characterization of solution set.

Vaccination for communicable endemic diseases: optimal allocation of initial and booster vaccine doses.

For some communicable endemic diseases (e.g., influenza, COVID-19), vaccination is an effective means of preventing the spread of infection and reducing mortality, but must be augmented over time with vaccine booster doses. We consider the problem of optimally allocating a limited supply of vaccines over time between different subgroups of a population and between initial versus booster vaccine doses, allowing for multiple booster doses. We first consider an SIS model with interacting population groups and four different objectives: those of minimizing cumulative infections, deaths, life years lost, or quality-adjusted life years lost due to death. We solve the problem sequentially: for each time period, we approximate the system dynamics using Taylor series expansions, and reduce the problem to a piecewise linear convex optimization problem for which we derive intuitive closed-form solutions. We then extend the analysis to the case of an SEIS model. In both cases vaccines are allocated to groups based on their priority order until the vaccine supply is exhausted. Numerical simulations show that our analytical solutions achieve results that are close to optimal with objective function values significantly better than would be obtained using simple allocation rules such as allocation proportional to population group size. In addition to being accurate and interpretable, the solutions are easy to implement in practice. Interpretable models are particularly important in public health decision making.

Algorithmic solutions for maximizing shareable costs

AbstractThis article addresses the linear optimization problem to maximize the total costs that can be shared among a group of agents, while maintaining stability in the sense of the core constraints of a cooperative transferable utility game, or TU game. When maximizing total shareable costs, the cost shares must satisfy all constraints that define the core of a TU game, except for being budget balanced. The article first gives a fairly complete picture of the computational complexity of this optimization problem, its relation to optimization over the core itself, and its equivalence to other, minimal core relaxations that have been proposed earlier. We then address minimum cost spanning tree (MST) games as an example for a class of cost sharing games with non‐empty core. While submodular cost functions yield efficient algorithms to maximize shareable costs, MST games have cost functions that are subadditive, but generally not submodular. Nevertheless, it is well known that cost shares in the core of MST games can be found efficiently. In contrast, we show that the maximization of shareable costs is ‐hard for MST games and derive a 2‐approximation algorithm. Our work opens several directions for future research.

Generating linear, semidefinite, and second-order cone optimization problems for numerical experiments

The numerical performance of algorithms can be studied using test sets or procedures that generate such problems. This paper proposes various methods for generating linear, semidefinite, and second-order cone optimization problems. Specifically, we are interested in problem instances requiring a known optimal solution, a known optimal partition, a specific interior solution, or all these together. In the proposed problem generators, different characteristics of optimization problems, including dimension, size, condition number, degeneracy, optimal partition, and sparsity, can be chosen to facilitate comprehensive computational experiments. We also develop procedures to generate instances with a maximally complementary optimal solution with a predetermined optimal partition to generate challenging semidefinite and second-order cone optimization problems. Generated instances enable us to evaluate efficient interior-point methods for conic optimization problems.

THE GOLDMAN-TUCKER THEOREM FOR TWO-SIDE CONSTRAINTS LINEAR OPTIMIZATION PROBLEM

The Goldman-Tucker theorem holds significant importance in linear optimization, as it ensures the existence of a strictly complementary solution. Two reasons underscore its relevance: firstly, logarithmic barrier interior point methods converge towards a strictly complementary solution, whose existence is warranted by this theorem. Secondly, the Goldman-Tucker theorem, coupled with the complementary slackness condition of the KKT (Karush–Kuhn–Tucker) system, has motived the development of efficient preconditioners for the final iterations of interior point methods, such as the Splitting preconditioner. Academic texts of linear optimization present this result as well as its consequences only for the canonical and standard form. This paper aims to accurately elucidate this theorem for the two-side constraints linear optimization problem with a detailed demonstration. Additionally, a theoretical result that uses this theorem to show that the central path converges to a strictly complementary solution and an example of application of both theoretical results are presented. Keywords: Interior point method; Central path; Strictly

Optimality conditions and duality for a class of continuous-time linear optimization problems

Optimality conditions and duality for a class of continuous-time linear optimization problems

Quantifying and reducing the complexity of multi-line charts as a visual aid in multi-criteria decision-making

AbstractMulti-line charts are commonly used in multi-criteria decision-making (MCDM) to represent multiple data series on the same graph. However, the presence of conflicting criteria or divergent viewpoints introduces the challenge of accurately interpreting these charts, necessitating thoughtful design to improve their comprehensibility. In this paper, we model these multi-line charts as connected perfect matching bipartite graphs. We propose a metric called the Coefficient of Complexity (CoC) to quantify the complexity of these multi-line charts. In order to reduce the visual complexity of these charts, we propose to minimize the CoC by modeling it as an integer linear optimization problem (reminiscent of the traveling salesman problem). We demonstrate our techniques through multiple real-life case studies, wherein multi-line charts serve as data visualization across various MCDM software tools. Additionally, multi-line charts with specific requirements have been optimized using our approach, showcasing the adaptability and efficacy of our technique. We also formulate the radar chart as a specialized form of the multi-line chart, and adapt our technique to improve its comprehensibility. The proposed CoC and its optimization are important contributions to the field of analytics, as a number of methods use multi-line charts for visual aid. Consequently, enhancing their comprehensibility can facilitate the decision-making process and help decision-makers gain insights.

A machine learning approach to two-stage adaptive robust optimization

We propose an approach based on machine learning to solve two-stage linear adaptive robust optimization (ARO) problems with binary here-and-now variables and polyhedral uncertainty sets. We encode the optimal here-and-now decisions, the worst-case scenarios associated with the optimal here-and-now decisions, and the optimal wait-and-see decisions into what we denote as the strategy. We solve multiple similar ARO instances in advance using the column and constraint generation algorithm and extract the optimal strategies to generate a training set. We train machine learning models that predict high-quality strategies for the here-and-now decisions, the worst-case scenarios associated with the optimal here-and-now decisions, and the wait-and-see decisions. The models can be applied to problems with varying dimensions. We also introduce novel methods to expedite training data generation and reduce the number of different target classes the machine learning algorithm needs to be trained on. We apply the proposed approach to the facility location, the multi-item inventory control and the unit commitment problems. Our approach solves ARO problems drastically faster than the state-of-the-art algorithms with high accuracy.