Causes Of Infeasibility Research Articles

Diagnosis of linear programming supply chain optimization models: Detecting infeasibilities and minimizing changes for new parameter values

In this paper, we address two major challenges in linear programming supply chain models: detecting infeasibilities and minimizing changes when new parameter data are introduced in the model. First, we address the problem of detecting and restoring feasibility using the flexibility test method to quantitatively evaluate the constraints in the model that are causing infeasibility. If the parameters in these infeasible constraints were incorrectly specified, we use regression and time-series models to detect outliers in the data that may be the cause of infeasibility. Corrective actions may be taken by the user upon identification of the cause of infeasibility through this algorithm. Second, we address the problem of minimizing changes in the solution of the linear programming model that are introduced due to varying parameters. The three formulations minimize the magnitude of changes, number of changes, and the weighted sum of both magnitude and number of changes in the model. We also formulate a bi-criterion optimization model to consider the objectives of minimizing cost and minimizing the weighted sum of the number and magnitude of changes to analyze the trade-off between the two objectives. An ideal compromise solution between the two objectives is also presented. The proposed algorithms are applied to supply chain problems including real industrial problems, to demonstrate their usefulness.

Infeasibility resolution for multi-purpose batch process scheduling

Scheduling decisions give rise to some of the most challenging optimization problems in the process industry. Formulating mathematical models for scheduling problems and devising tailored solution algorithms for these models has been the main thrust of previous literature. In this work, we focus on analyzing and resolving the cause of bottlenecks and infeasibilities in generic scheduling problems. We present a systematic approach for infeasibility diagnosis. Our approach exploits the known structure of scheduling models to isolate interpretable infeasible sets of constraints. We demonstrate the power of the algorithm on infeasible instances of the Westenberger–Kallrath multipurpose batch process modeled using a state-task network (STN) representation. The methodology presented in the paper is able to successfully analyze the cause of infeasibility and provide recommendations for resolving it. We also demonstrate how these insights and recommendations can be presented to scheduling operators with little or no optimization expertise in an intuitive manner.

Composed IIS path locating based optimization for bounded \\reachability analysis of compositional linear hybrid systems}{Composed IIS path locating based optimization for bounded reachability analysis of compositional linear hybrid systems

Hybrid systems include both discrete and continuous behavior and are widely used to model control systems. The reachability analysis of its unsafe state is an important method for guaranteeing the safety of a system. However, the current techniques do not scale well to the problems of practical interest. Due to the synchronization of components and the combinatorial explosion of state space, the reachability analysis of compositional linear hybrid system is extremely complex. In order to reduce the complexity, a path-oriented approach was proposed in a previous work, which conducted bounded reachability analysis of a compositional linear hybrid system. By enumerating and verifying each potential path one by one, the size of the problem that can be solved will be increased substantially. This path-oriented approach will become quite inefficient due to a sharp increase in the number of candidate paths when analyzing complex systems. The path explosion problem in model checking is also famous. To solve this problem, we propose a state-space reduction technique, which accelerates the verification process. We propose a method to locate the cause of infeasibility, when a composed infeasible path segment after a path set is proved to be infeasible. As we can simply falsify a path set that contains a composed infeasible path segment, the number of candidate paths can be reduced significantly. Furthermore, to avoid such composed path segments efficiently, we propose an approach based on satisfiability modulo theories (SMT), to traverse the bounded graph structure of the composed linear hybrid system. The results of the experiment show that the performance of the path-oriented bounded reachability analysis can be optimized significantly and that the overall performance of the proposed approach is better than that of the state-of-the-art competitor.

Analysis of Infeasible Cases in Optimal Power Flow Problem

Abstract: In the context of smart grid transformation of existing electricity networks, optimal power flow (OPF) and security-constrained OPF (SCOPF) studies remain to be very important for power system planning, operation and market analysis. OPF study involves finding the (global) optimum solution to a set of nonlinear algebraic equations, subjected to a set of equality and inequality constraints. When the system is heavily stressed, particularly following a severe contingency, the conventional OPF methods may fail due to the problem or solution infeasibility, or inability to select proper initial values. The soft constraint handling approach and repetitive constraint relaxation in finding the causes of infeasibility could be either tedious, or may not be practical for the large-scale problems. This paper presents an alternative approach, based on use of meta-heuristic method, to pinpoint the main reasons for the failure of solution algorithms in nonlinear optimization, in general, and OPF problem, in particular. The presented approach is illustrated on commonly used IEEE 14-bus and 30-bus test networks.

Extracting minimal unsatisfiable subformulas in satisfiability modulo theories

Explaining the causes of infeasibility of formulas has practical applications in various fields, such as formal verification and electronic design automation. A minimal unsatisfiable subformula provides a succinct explanation of infeasibility and is valuable for applications. The problem of deriving minimal unsatisfiable cores from Boolean formulas has been addressed rather frequently in recent years. However little attention has been concentrated on extraction of unsatisfiable subformulas in Satisfiability Modulo Theories(SMT). In this paper, we propose a depth-firstsearch algorithm and a breadth-first-search algorithm to compute minimal unsatisfiable cores in SMT, adopting different searching strategy. We report and analyze experimental results obtaining from a very extensive test on SMT-LIB benchmarks.

Pseudo-loadflow formulation as a starting process for the Newton Raphson algorithm

This paper introduces new models which approximate the AC loadflow problem, but are able to converge (using the Newton Raphson algorithm) from a wider range of starting points. The solution of the pseudo-loadflow models can provide a robust starting process for the Newton Raphson solution of the conventional loadflow problem. It is also shown that pseudo-loadflow solutions exist in many cases where the AC loadflow equations do not appear to have any solution, and in such cases the pseudo-loadflow solution can provide useful information to assist in locating the cause of infeasibility of the AC loadflow model. Test results are presented for illustrative small network examples and also for larger test networks. The computational requirements of the proposed methods are similar to those of the conventional Newton Raphson loadflow algorithm.

A goal programming approach for farm planning with resources dimensionality

Crop production entails many decision making processes aimed at improving productivity and achieving the best yield from scarce resources, which are normally limited. Assuming that there is a certain technical path of tasks to be carried out within a period, and that each task can be done in different ways, the problem addressed in this paper consists of choosing how and when to carry out each one, in such a way that the tasks are scheduled in sequence at the lowest possible cost, taking account of any relations of precedence among them, and in such a way that each task is done within its time window and with the resources being assigned in a feasible way. Time windows are usually defined in a strict way, and thus, if adjusted, can be a cause of infeasibility for some of the problems, implying the need to acquire new resources. Nevertheless, in most cases small deviations from time windows could be acceptable. In this paper a mixed 0-1 programming model to attain the proposed objective is proposed, applying goal programming to model time windows in a more flexible way.

Tracking Unsatisfiable Subformulas from Reduced Refutation Proof

Explaining the causes of infeasibility of Boolean formulas has many practical applications in various fields. A small unsatisfiable subformula provides a succinct explanation of infeasibility and is valuable for applications. In recent years finding unsatisfiable subformulas has been addressed frequently by research works, mostly based on the SAT solvers with DPLL backtrack-search algorithm. However little attention has been concentrated on extraction of unsatisfiable subformulas using incomplete methods. In this paper, we present the definitions of refutation proof and refutation parsing graph, and then propose a resolution based local search algorithm to track unsatisfiable subformulas according to the reduced refutation proof of a formula. This approach directly constructs the resolution sequences for proving unsatisfiability with a local search procedure, and then recursively derives unsatisfiable subformulas from the resolve traces. We report and analyze the experimental results on well-known and randomly generated benchmarks.

An ant colony algorithm for minimum unsatisfiable core extraction

Explaining the causes of infeasibility of Boolean formulas has many practical applications in electronic design automation and formal verification of hardware. Furthermore, a minimum explanation of infeasibility that excludes all irrelevant information is generally of interest. A smallest-cardinality unsatisfiable subset called a minimum unsatisfiable core can provide a succinct explanation of infeasibility and is valuable for applications. However, little attention has been concentrated on extraction of minimum unsatisfiable core. In this paper, the relationship between maximal satisfiability and minimum unsatisfiability is presented and proved, then an efficient ant colony algorithm is proposed to derive an exact or nearly exact minimum unsatisfiable core based on the relationship. Finally, experimental results on practical benchmarks compared with the best known approach are reported, and the results show that the ant colony algorithm strongly outperforms the best previous algorithm.

A branch and bound algorithm for extracting smallest minimal unsatisfiable subformulas

Explaining the causes of infeasibility of Boolean formulas has practical applications in numerous fields, such as artificial intelligence (repairing inconsistent knowledge bases), formal verification (abstraction refinement and unbounded model checking), and electronic design (diagnosing and correcting infeasibility). Minimal unsatisfiable subformulas (MUSes) provide useful insights into the causes of infeasibility. An unsatisfiable formula often has many MUSes. Based on the application domain, however, MUSes with specific properties might be of interest. In this paper, we tackle the problem of finding a smallest-cardinality MUS (SMUS) of a given formula. An SMUS provides a succinct explanation of infeasibility and is valuable for applications that are heavily affected by the size of the explanation. We present (1) a baseline algorithm for finding an SMUS, founded on earlier work for finding all MUSes, and (2) a new branch-and-bound algorithm called Digger that computes a strong lower bound on the size of an SMUS and splits the problem into more tractable subformulas in a recursive search tree. Using two benchmark suites, we experimentally compare Digger to the baseline algorithm and to an existing incomplete genetic algorithm approach. Digger is shown to be faster in nearly all cases. It is also able to solve far more instances within a given runtime limit than either of the other approaches.

Solving Infeasibility Problems in Computerized Test Assembly

Linear programming techniques have been used successfully in a variety of test assembly problems. It is, however, possible that no test can be found meeting all the constraints in the linear programming model. The problem of diagnosing and repairing infeasible linear programming models is discussed. It is demonstrated that it is possible to localize the causes of infeasibility.