Windy Postman Problem Under Uncertainty

The windy postman problem is the NP-hard problem of finding the minimum cost of a tour traversing all edges of an undirected graph, where the cost of traversal of an edge depends on the direction. Given an undirected graph $G$, we consider the polyhedron $O(G)$ induced by the linear programming relaxation of a well-known integer programming formulation of the problem. We say that $G$ is windy postman perfect if $O(G)$ is integral. There exists a polynomial-time algorithm, based on the ellipsoid method, to solve the windy postman problem for the class of windy postman perfect graphs. Eulerian graphs and trees are windy postman perfect. By considering a family of polyhedra related to $O(G)$, we prove that series-parallel graphs are windy postman perfect, therefore solving a conjecture of [Win1987a].

On the use of uncertainty analyses to test hypotheses regarding deterministic model predictions of environmental processes

This paper presents optimality criteria for fuzzy-valued fractional multi-objective optimization problem. There are numerous optimality criteria which have been established for the deterministic fractional multi-objective optimization problems. Very few studies are available on the establishment of optimality criteria for fuzzy-valued multi-objective optimization problem. So, Karush–Kuhn–Tucker optimality criteria for fuzzy-valued fractional multi-objective problem are established by using Lagrange multipliers. First, the original problem is modified using the parametric approach of Dinkelbach into multi-objective non-fractional optimization problem, and then, the optimality conditions are established for the modified problem using the Hukuhara derivative. The established optimality criteria are verified by two numerical examples.

This paper considers robot motion planning under temporal logic constraints in probabilistic maps obtained by semantic simultaneous localization and mapping (SLAM). The uncertainty in a map distribution presents a great challenge for obtaining correctness guarantees with respect to the linear temporal logic (LTL) specification. We show that the problem can be formulated as an optimal control problem in which both the semantic map and the logic formula evaluation are stochastic. Our first contribution is to reduce the stochastic control problem for a subclass of LTL to a deterministic shortest path problem by introducing a confidence parameter $delta$. A robot trajectory obtained from the deterministic problem is guaranteed to have minimum cost and to satisfy the logic specification in the true environment with probability $delta$. Our second contribution is to design an admissible heuristic function that guides the planning in the deterministic problem towards satisfying the temporal logic specification. This allows us to obtain an optimal and very efficient solution using the A* algorithm. The performance and correctness of our approach are demonstrated in a simulated semantic environment using a differential-drive robot.

Because of the complexity of some optimization problems, evolutionary and meta-heuristic algorithms are sometimes more applicable than the traditional optimization methods. Some difficulties in solving design-operation problems in the field of engineering are due to the multi-modality of the solution region of these problems. Since the design variables usually are specified as discrete variables and other continuous decision variables have to be set according to the range of the discrete ones, the possibility of trapping the final solution into some local optimum increases. In such cases, the capability of both traditional and evolutionary algorithms decreases. Thus, the development of a strategy to overcome this problem is the subject of this paper. For water utilities, one of the greatest potential areas for energy cost-savings is the effective scheduling of daily pump operations. Optimum design operation of pumping stations is a potential problem in this area that performs a wide background of solutions to this problem with different methods. Computation in all methods is driven by an objective function that includes operating and capital costs subject to various performances and hydraulic constraints. This paper achieves the optimal control and operation of an irrigation pumping station system by one of the latest tools used in optimization problems, which is the honey-bees mating optimization (HBMO) algorithm and is tested with a practical design. The HBMO algorithm with dynamic penalty function is presented and compared with two other well-known optimization tools which are the Lagrange multipliers (LM) method and genetic algorithms (GA) as well as with the previous results of the HBMO algorithm with constant penalty function for the same problem. The LM, GA and HBMO approaches simultaneously determine the least total annual cost of the pumping station and its operation. The solution includes the selection of pump type, capacity and the number of units, as well as scheduling the operation of irrigation pumps that results in minimum design and operating cost for a set of water demand curves. In this paper, the HBMO algorithm is applied and the dynamic penalty function is tested to demonstrate the efficiency of this combination simultaneously. The results are very promising and prove the ability of combining the dynamic penalty function with the HBMO algorithm for solving combinatorial design–operation optimization problems. Application of all these models to a real-world project shows not only considerable savings in cost and energy but also highlights the efficiency and capability of the dynamic penalty function in the HBMO algorithm for solving complex problems of this type.

This study proposes an on-line minimum loss (ML) control strategy that considers the iron and copper losses of an interior permanent magnet synchronous motor in torque-controlled applications. The proposed ML control strategy utilizes a constrained optimization problem to satisfy torque reference tracking and loss minimization. An equivalent iron loss resistance circuit is adopted for the loss minimization. To solve the optimization problem, the Lagrange multiplier method is applied through a numerical analysis algorithm. The resulting solution provides the optimal current reference for every sampling instant. The Lagrange multiplier method needs parameters such as magnetic flux linkages, dynamic inductances, and iron loss resistance. Fundamental magnetic flux linkages are estimated using a flux observer, and the dynamic inductances are estimated with high-frequency voltage signal injection. The proposed iron loss resistance estimator estimates the equivalent iron resistance without any preliminary experiments. DC input power is measured using a current sensor for the accurate on-line estimation of iron loss resistance. To analyze the effectiveness of ML control compared with conventional maximum torque per ampere control, finite element analysis is used. The feasibility of the proposed ML control strategy is verified through simulation and experiments.

A Lagrange barrier approach for the minimum concave cost supply problem via a logarithmic descent direction algorithm

The objective in this paper is about economic dispatch problem of electric power generation where scheduling the committed generating units outputs so as to meet the required load demand at minimum operating cost, while satisfying all units and system equality and inequality constraint. In the operating of electric power system, an economic planning problem is one of variables that its must be considered since economically planning will give more efficiency in operational cost. In this paper the economic dispatch problem which has non linear cost function solved by using swarm intelligent method is Gaussian Particle Swarm Optimization (GPSO) and Lagrange Multiplier. GPSO is a population-based stochastic algorithms which their moving inspired by swarm intelligent and probabilities theories. To analize its accuracy, the economic dispatch solution by GPSO method will be compared with Lagrange multiplier method. From the running test result the GPSO method give economically planning calculation which it better than Lagrange multiplier method and the GPSO method faster to getting error convergence. Therefore the GPSO method have better performance to getting global best solution than the Lagrange method.

A minimum cost production model which can be used to account economical energy exchange between utilities and evaluate production cost saving in multiple area interconnected systems is proposed in this paper. A fast probabilistic production simulation model is used to determine the thermal incremental cost of each system separately. Based on the thermal incremental cost of each subsystem, a nonlinear optimization model which minimizes the total production cost of the whole interconnected systems is used to determine the economical energy exchanges in the interconnected systems. Then total production cost and cost saving are for whole interconnected systems and each system. The minimum cost principle based on Lagrange multipliers is proposed to solve this nonlinear optimization problem. The tie line capacity limit and energy loss are included in the energy exchange process. Hourly loads are used in the model. The thermal incremental cost of each system and the energy exchanges between interconnected systems are evaluated hour by hour. Then a period summary report is produced. The proposed minimum cost model has been incorporated into FEPSS (Fast Electricity Production Simulation System) package. The Pacific Power Corporation, Australia has used this model to study power system interconnections in Southeast Australia. >

This work introduces the concept of deterministic annealing (DA) as a useful approach to clustering and other related optimization problems. It is strongly motivated by analogies to statistical physics, but is formally derived within information theory and probability theory. This approach enables escaping local optima that plague traditional techniques, without the extremely slow schedules typically required by stochastic methods. The clustering solutions obtained by DA are totally independent of the choice of initial configuration. A probabilistic framework is constructed, which is based on the principle of maximum entropy. The association probabilities at a given average distortion are Gibbs distributions parametrized by the corresponding Lagrange multiplier $\beta$, which is inversely proportional to the temperature in the physical analogy. By computing marginal probabilities, an effective cost is obtained, which is minimized to find the most probable set of cluster representatives at a given temperature. This effective cost is the free energy in statistical mechanics, which is indeed optimized at isothermal, stochastic equilibrium. Within the probabilistic framework, annealing is introduced by controlling the Lagrange multiplier $\beta$. This annealing is interpreted as gradually reducing the fuzziness of the associations. Phase transitions are identified in the process, which are, in fact, cluster splits. A sequence of phase transitions produces a hierarchy of fuzzy-clustering solutions. Critical $\beta$ are computed exactly for the first phase transition and approximately for the following ones. Specific algorithms are derivable from the general approach, to address different aspects of clustering. From the experimental results it appears that DA is substantially superior to traditional techniques. The approach is extended to deal with a larger family of optimization problems that can be reformulated as constrained clustering. A probabilistic framework for constrained clustering is derived. Three examples are discussed. Mass-constrained clustering yields an improvement of the clustering procedure. The process is now independent of the number and multiplicities of representatives. The travelling salesman problem is reformulated as constrained clustering, yielding the elastic net approach. A second Lagrange multiplier is identified and is controlled in the process. Finally, this approach is suggested for self-organization of neural networks.

SummaryWe consider optimization problems with inequality and abstract set constraints, and we derive sensitivity properties of Lagrange multipliers under very weak conditions. In particular, we do not assume uniqueness of a Lagrange multiplier or continuity of the perturbation function. We show that the Lagrange multiplier of minimum norm defines the optimal rate of improvement of the cost per unit constraint violation.

The Schur complement trick appears sporadically in numerical optimization methods [Schur 1917; Cottle 1974]. The trick is especially useful for solving Lagrangian saddle point problems when minimizing quadratic energies subject to linear equality constraints [Gill et al. 1987]. Typically, to apply the trick, the energy's Hessian is assumed positive definite. I generalize this technique for positive semi-definite Hessians.

Optimization problems are typically solved by starting with an initial estimate and proceeding iteratively to improve it until the optimum is found. The design points along the path from the initial estimate to the optimum are usually of no value. The present work proposes a strategy for tracing a path of optimum solutions parameterized by the amount of available resources. The paper specifically treats the optimum design of a structure to maximize its buckling load. Equations for the optimum path are obtained using Lagrange multipliers, and solved by a homotopy method. The solution path has several transitions from unimodal to bimodal solutions. The Lagrange multipliers and second-order optimality conditions are used to detect branching points and to switch to the optimum solution path. The procedure is applied to the design of a foundation which supports a column for maximum buckling load. Using the total available foundation stiffness as a homotopy parameter, a set of optimum foundation designs is obtained.

Constrained optimization problems are very important as they are encountered in many science and engineering applications. A hybrid method based on modified augmented Lagran- gian multiplier and biogeography-based optimization (BBO) algorithm is proposed to solve con- strained optimization problems. The basic steps of the proposed method are comprised of an outer iteration, in which the Lagrangian multipliers and various penalty parameters are updated using a first-order update scheme, and an inner iteration, in which a nonlinear optimization of the modified augmented Lagrangian function with simple bound constraints is implemented by BBO algorithm. Numerical results show that the proposed method is reliable and efficient for solving constrained optimization problems.

Relevance. The need to improve the reliability and efficiency of the technological ore grinding system by element-by-element optimal loaded redundancy of its elements. Aim. To substantiate the need to improve the efficiency of the ore grinding technological system functioning by optimal redundancy; to find out an approximate solution of the following problems: 1) to maximize the efficiency of the ore grinding complex system functioning by element-by-element loaded redundancy limited, superimposed on the technical and economic indicator of the system; 2) to minimize technical economic indicator of the ore grinding system at the set value of the efficiency indicator of its functioning. Object. Ore grinding technological system, considered as a system with many operable states (multi-state system), the elements of which have only two possible states. The system under consideration is reserved by the element-by-element loaded redundancy method. Methods. Method for evaluating the efficiency of functioning of the systems with many states, the method of indefinite Lagrange multipliers. Results. The authors have obtained an approximate value of the performance indicator of the ore grinding technological system with many workable states, the elements of which have only two possible states. The paper considers the problems of optimal redundancy of the ore grinding complex technological system. Approximate solutions of these problems are obtained by the method of indefinite Lagrange multipliers, which can be used in rough calculations at the stage of designing complex technological systems for grinding ore. The problem of optimal redundancy is also considered, when it is possible to reserve only m elements from the n elements of ore grinding complex system. Taking into account certain assumptions for the efficiency of the system functioning, reserved by the element-by-element loaded redundancy method, an approximate solution of this problem is obtained.