Browsing by Subject "Linear programming"

Now showing 1 - 20 of 49

Open Access
An algorithm based on facial decomposition for finding the efficient set in multiple objective linear programming
(Elsevier, 1996) Sayın, S.
We propose a method for finding the efficient set of a multiple objective linear program based on the well-known facial decomposition of the efficient set. The method incorporates a simple linear programming test that identifies efficient faces while employing a top-down search strategy which avoids enumeration of efficient extreme points and locates the maximally efficient faces of the feasible region. We suggest that discrete representations of the efficient faces could be obtained and presented to the Decision Maker. Results of computational experiments are reported.
Open Access
Auction design and optimal allocation by linear programming
(2015) Bayrak, Halil İbrahim
For the sale of a single object through an auction, we assume discrete type space for agents and make use of linear programming to find optimal mechanism design for a risk-neutral seller. First, we show that the celebrated incentive compatible mechanism, second price auction, is not optimal. We find a slightly different optimal mechanism referred to as “discrete second price auction”. Second we consider the problem of allocation with costly inspection. We obtain the optimal solution in the form of a favored-agent mechanism by the Greedy Algorithm. Moreover, we relax the common prior assumption and maximize the worst-case utility of an ambiguity averse seller for the two problems mentioned above. While the problem does not yield a useful optimal mechanism in general, optimal solutions for some special cases are obtained.
Open Access
A Benders decomposition approach for the charging station location problem with plug-in hybrid electric vehicles
(Elsevier, 2016) Arslan, O.; Karaşan, O. E.
The flow refueling location problem (FRLP) locates p stations in order to maximize the flow volume that can be accommodated in a road network respecting the range limitations of the vehicles. This paper introduces the charging station location problem with plug-in hybrid electric vehicles (CSLP-PHEV) as a generalization of the FRLP. We consider not only the electric vehicles but also the plug-in hybrid electric vehicles when locating the stations. Furthermore, we accommodate multiple types of these vehicles with different ranges. Our objective is to maximize the vehicle-miles-traveled using electricity and thereby minimize the total cost of transportation under the existing cost structure between electricity and gasoline. This is also indirectly equivalent to maximizing the environmental benefits. We present an arc-cover formulation and a Benders decomposition algorithm as exact solution methodologies to solve the CSLP-PHEV. The decomposition algorithm is accelerated using Pareto-optimal cut generation schemes. The structure of the formulation allows us to construct the subproblem solutions, dual solutions and nondominated Pareto-optimal cuts as closed form expressions without having to solve any linear programs. This increases the efficiency of the decomposition algorithm by orders of magnitude and the results of the computational studies show that the proposed algorithm both accelerates the solution process and effectively handles instances of realistic size for both CSLP-PHEV and FRLP.
Open Access
Bicriteria multiresource generalized assignment problem
(John Wiley & Sons, 2014) Karsu, O.; Azizoglu, M.
In this study,we consider a bicriteria multiresource generalized assignment problem. Our criteria are the total assignment load and maximum assignment load over all agents. We aim to generate all nondominated objective vectors and the corresponding efficient solutions. We propose several lower and upper bounds and use them in our optimization and heuristic algorithms. The computational results have shown the satisfactory behaviors of our approaches.
Open Access
Bilateral trade in discrete type spaces by linear and convex optimization with application to supply chain coordination
(INSA Lyon, 2018) Pinar, M. C.
We consider the bilateral trade of an object between a seller and a buyer through an intermediary who aims to maximize his expected gains as in Myerson and Satterthwaite [1], in a Bayes-Nash equilibrium framework where the seller and buyer have private, discrete valuations for the object. Using duality of linear network optimization, the intermediary's initial problem is transformed into an equivalent linear programming problem with explicit formulae of expected revenues of the seller and the expected payments of the buyer, from which the optimal mechanism is immediately obtained. Then, an extension due to Spulber [2] is considered where the seller is also a producer with a cost parameter that is private information. Assuming suitable utility and production functions for the respective parties, the resulting non-convex mechanism design problem of a utility maximizing broker is revealed to have hidden convexity and transformed into an equivalent (almost) unconstrained convex optimization problem over output variables, which, in many cases, can be solved easily by calculus. A contracting game under asymmetric information specific to two-echelon supply chain coordination between a retailer of unknown type and a supplier, akin to the bilateral trade game of the paper, is also studied. Special cases leading to closed-form solution with increasing information rent for higher types are identified along with the requisite conditions for their validity.
Open Access
Birleşik sezim ve kestirim sistemlerinin gürültü ile geliştirilmesi
(IEEE, 2014-04) Akbay, Abdullah Başar; Gezici, Sinan
Belirli koşullar altında, optimal olmayan bazı sezici ve kestiricilerin performansını girdilerine gürültü ekleyerek geliştirmek mümkündür. Bu çalışmada, birleşik bir sezim ve kestirim sisteminin gürültü eklenerek geliştirilmesi incelenmektedir. Sistem performansının maksimizasyonu bir optimizasyon problemi olarak tanımlanmaktadır. Optimal toplanır gürültü dağılımının istatiksel özellikleri belirlenmektedir. Sistem performansının gürültü ile iyileştirilemeyeceği bir koşul elde edilmektedir.Önerilen optimizasyon probleminin, bir doğrusal programlama (DP) problemi olarak yaklaşımı sunulmaktadır. Bir sayısal örnek üzerinde, kuramsal bulguları desteklemek amacıyla, gürültü eklenmiş sistem ile orijinal sistemin performansları karşılaştırılmaktadır.
Open Access
A characterization of the optimal set of linear programs based on the augmented lagrangian
(Taylor & Francis, 1999) Pınar, M. Ç.
It is proved that in a certain neighborhood of the optimal set of multipliers, the set of minimizers of the augmented lagrangian nmction generates a new characterization of the optimal solution set of the linear program.
Open Access
Concurrent consideration of product design, process planning and production planning activities
(1998) Adil, Gajendra Kumar
In manufacturing engineering, product design, process planning and production planning activities are often considered independently. However, in order to effectively respond to changes in business situations, such as changes in demand forecast, product mix and technology, it is desirable to consider them concurrently. For this purpose, a large-scale linear programming model has been developed. The model considers minimization of the sum of processing cost, late shipment cost and inventory holding cost as the objective, and concurrently selects product designs, and generates process plans and production plans. The number of columns in the formulation can be large and, hence, an efficient column generation scheme is developed to solve the model. The model and solution procedure are illustrated with examples.
Open Access
Continuous knapsack sets with divisible capacities
(Springer, 2016) Wolsey, L. A.; Yaman, H.
We study two continuous knapsack sets (Formula presented.) and (Formula presented.) with (Formula presented.) integer, one unbounded continuous and (Formula presented.) bounded continuous variables in either (Formula presented.) or (Formula presented.) form. When the coefficients of the integer variables are integer and divisible, we show in both cases that the convex hull is the intersection of the bound constraints and (Formula presented.) polyhedra arising as the convex hulls of continuous knapsack sets with a single unbounded continuous variable. The latter convex hulls are completely described by an exponential family of partition inequalities and a polynomial size extended formulation is known in the (Formula presented.) case. We also provide an extended formulation for the (Formula presented.) case. It follows that, given a specific objective function, optimization over both (Formula presented.) and (Formula presented.) can be carried out by solving (Formula presented.) polynomial size linear programs. A further consequence of these results is that the coefficients of the continuous variables all take the values 0 or 1 (after scaling) in any non-trivial facet-defining inequality of the convex hull of such sets.
Open Access
Decomposing linear programs for parallel solution
(Springer, 1995-08) Pınar, Ali; Çatalyürek Ümit V.; Aykanat, Cevdet; Pınar, Mustafa
Coarse grain parallelism inherent in the solution of Linear Programming (LP) problems with block angular constraint matrices has been exploited in recent research works. However, these approaches suffer from unscalability and load imbalance since they exploit only the existing block angular structure of the LP constraint matrix. In this paper, we consider decomposing LP constraint matrices to obtain block angular structures with specified number of blocks for scalable parallelization. We propose hypergraph models to represent LP constraint matrices for decomposition. In these models, the decomposition problem reduces to the well-known hypergraph partitioning problem. A Kernighan-Lin based multiway hypergraph partitioning heuristic is implemented for experimenting with the performance of the proposed hypergraph models on the decomposition of the LP problems selected from NETLIB suite. Initial results are promising and justify further research on other hypergraph partitioning heuristics for decomposing large LP problems. © Springer-Verlag Berlin Heidelberg 1996.
Open Access
A derivation of Lovász' theta via augmented lagrange duality
(E D P Sciences, 2003) Pınar, M. Ç.
A recently introduced dualization technique for binary linear programs with equality constraints, essentially due to Poljak et al. [13], and further developed in Lemar´echal and Oustry [9], leads to simple alternative derivations of well-known, important relaxations to two well-known problems of discrete optimization: the maximum stable set problem and the maximum vertex cover problem. The resulting relaxation is easily transformed to the well-known Lov´asz θ number.
Open Access
Design of translucent optical networks: Partitioning and restoration
(Kluwer, 2004) Karasan, E.; Arisoylu, M.
We discuss the problem of designing translucent optical networks composed of restorable, transparent subnetworks interconnected via transponders. We develop an integer linear programming (ILP) formulation for partitioning an optical network topology into subnetworks, where the subnetworks are determined subject to the constraints that each subnetwork satisfies size limitations, and it is two-connected. A greedy heuristic partitioning algorithm is proposed for planar network topologies. We use section restoration for translucent networks where failed connections are rerouted within the subnetwork which contains the failed link. The network design problem of determining working and restoration capacities with section restoration is formulated as an ILP problem. Numerical results show that fiber costs with section restoration are close to those with path restoration for mesh topologies used in this study. It is also shown that the number of transponders with the translucent network architecture is substantially reduced compared to opaque networks.
Open Access
Discrete-time pricing and optimal exercise of American perpetual warrants in the geometric random walk model
(2013) Vanderbei, R. J.; Pınar, M. Ç.; Bozkaya, E. B.
An American option (or, warrant) is the right, but not the obligation, to purchase or sell an underlying equity at any time up to a predetermined expiration date for a predetermined amount. A perpetual American option differs from a plain American option in that it does not expire. In this study, we solve the optimal stopping problem of a perpetual American option (both call and put) in discrete time using linear programming duality. Under the assumption that the underlying stock price follows a discrete time and discrete state Markov process, namely a geometric random walk, we formulate the pricing problem as an infinite dimensional linear programming (LP) problem using the excessive-majorant property of the value function. This formulation allows us to solve complementary slackness conditions in closed-form, revealing an optimal stopping strategy which highlights the set of stock-prices where the option should be exercised. The analysis for the call option reveals that such a critical value exists only in some cases, depending on a combination of state-transition probabilities and the economic discount factor (i.e., the prevailing interest rate) whereas it ceases to be an issue for the put.
Open Access
An effective model to decompose linear programs for parallel solution
(Springer, 1996-08) Pınar, Ali; Aykanat, Cevdet
Although inherent parallelism in the solution of block angulax Linear Programming (LP) problems has been exploited in many research works, the literature that addresses decomposing constraint matrices into block angular form for parallel solution is very rare and recent. We have previously proposed hypergraph models, which reduced the problem to the hypergraph partitioning problem. However, the quality of the results reported were limited due to the hypergraph partitioning tools we have used. Very recently, multilevel graph partitioning heuristics have been proposed leading to very successful graph partitioning tools; Chaco and Metis. In this paper, we propose an effective graph model to decompose matrices into block angular form, which reduces the problem to the well-known graph partitioning by vertex separator problem. We have experimented the validity of our proposed model with various LP problems selected from NETLIB and other sources. The results are very attractive both in terms of solution quality and running times. © Springer-Verlag Berlin Heidelberg 1996.
Open Access
Equiripple FIR filter design by the FFT algorithm
(Institute of Electrical and Electronics Engineers, 1997-03) Çetin, A. Enis; Gerek, Ö. N.; Yardımcı, Y.
The fast Fourier transform (FFT) algorithm has been used in a variety of applications in signal and image processing. In this article, a simple procedure for designing finite-extent impulse response (FIR) discrete-time filters using the FFT algorithm is described. The zero-phase (or linear phase) FIR filter design problem is formulated to alternately satisfy the frequency domain constraints on the magnitude response bounds and time domain constraints on the impulse response support. The design scheme is iterative in which each iteration requires two FFT computations. The resultant filter is an equiripple approximation to the desired frequency response. The main advantage of the FFT-based design method is its implementational simplicity and versatility. Furthermore, the way the algorithm works is intuitive and any additional constraint can be incorporated in the iterations, as long as the convexity property of the overall operations is preserved. In one-dimensional cases, the most widely used equiripple FIR filter design algorithm is the Parks-McClellan algorithm (1972). This algorithm is based on linear programming, and it is computationally efficient. However, it cannot be generalized to higher dimensions. Extension of our design method to higher dimensions is straightforward. In this case two multidimensional FFT computations are needed in each iteration.
Open Access
Escaping local optima in a class of multi-agent distributed optimization problems: a boosting function approach
(IEEE, 2014) Sun, X.; Cassandras, C. G.; Gökbayrak, Kaan
We address the problem of multiple local optima commonly arising in optimization problems for multi-agent systems, where objective functions are nonlinear and nonconvex. For the class of coverage control problems, we propose a systematic approach for escaping a local optimum, rather than randomly perturbing controllable variables away from it. We show that the objective function for these problems can be decomposed to facilitate the evaluation of the local partial derivative of each node in the system and to provide insights into its structure. This structure is exploited by defining 'boosting functions' applied to the aforementioned local partial derivative at an equilibrium point where its value is zero so as to transform it in a way that induces nodes to explore poorly covered areas of the mission space until a new equilibrium point is reached. The proposed boosting process ensures that, at its conclusion, the objective function is no worse than its pre-boosting value. However, the global optima cannot be guaranteed. We define three families of boosting functions with different properties and provide simulation results illustrating how this approach improves the solutions obtained for this class of distributed optimization problems.
Open Access
Expected gain-loss pricing and hedging of contingent claims in incomplete markets by linear programming
(Elsevier, 2010) Pınar, M. Ç.; Salih, A.; Camcı, A.
We analyze the problem of pricing and hedging contingent claims in the multi-period, discrete time, discrete state case using the concept of a "λ gain-loss ratio opportunity". Pricing results somewhat different from, but reminiscent of, the arbitrage pricing theorems of mathematical finance are obtained. Our analysis provides tighter price bounds on the contingent claim in an incomplete market, which may converge to a unique price for a specific value of a gain-loss preference parameter imposed by the market while the hedging policies may be different for different sides of the same trade. The results are obtained in the simpler framework of stochastic linear programming in a multi-period setting, and have the appealing feature of being very simple to derive and to articulate even for the non-specialist. They also extend to markets with transaction costs.
Open Access
Formulations and valid inequalities for the heterogeneous vehicle routing problem
(Springer, 2006) Yaman, H.
We consider the vehicle routing problem where one can choose among vehicles with different costs and capacities to serve the trips. We develop six different formulations: the first four based on Miller-Tucker-Zemlin constraints and the last two based on flows. We compare the linear programming bounds of these formulations. We derive valid inequalities and lift some of the constraints to improve the lower bounds. We generalize and strengthen subtour elimination and generalized large multistar inequalities.
Open Access
Generalized column generation for linear programming
(Institute for Operations Research and the Management Sciences (INFORMS), 2002) Oğuz, O.
Column generation is a well-known and widely practiced technique for solving linear programs with too many variables or constraints to include in the initial formulation explicitly. Instead, the required column information is generated at each iteration of the simplex algorithm. This paper shows that, even if the number of variables is low enough for explicit inclusion in the model with the available technology, it may still be more efficient to resort to column generation for some class of problems.
Open Access
Generalized second price auction is optimal for discrete types
(Elsevier, 2016) Bayrak, H. I.; Pınar, M. Ç.
We prove that a variant of the second price auction for the sale of a single good through a Bayesian incentive compatible mechanism that maximizes expected revenue of the seller is optimal when the type space is discrete. Moreover, we show that this variant is related to the widely used generalized second price auction mechanism in keyword-auctions for advertising, thus providing a theoretical justification for a practical tool.