Multi-parametric programming based algorithms for the global solution of bi-level mixed-integer linear and quadratic programming problems

In this paper, we propose an alternate modified solution procedure based on fuzzy goal programming (FGP) approach for solving bi-level multiobjective linear fractional programming problems (BL-MOLFPPs) with some major modifications in the algorithm suggested by Abo-Sinna and Baky (2010). The proposed modified technique is simpler and requires less computational works than that of earlier technique. In the formulation of FGP models, each objective function (at both the levels - leader's level and follower's level) as well as control variables (of leader's level) are transformed into fuzzy goals with modified membership functions. Then FGP approach is used for achieving the highest degree of each of membership goals. To avoid decision deadlock situations and difficulties associated with bi-level programming problems, solution preferences by decision makers (at leader's and follower's levels) are not taken into account in proposed algorithm. A comparative analysis is also carried out with numerical example in order to show efficiency of proposed algorithm over earlier technique.

This article considers bilevel linear programming problems where the coefficients of the objective functions and the constraints in the problem are given as fuzzy parameters. Stackelberg problems and Stackelberg solutions under fuzziness have not been fully discussed so far. New Stackelberg solution concepts under fuzziness, called possibilistic Stackelberg solutions , are proposed by incorporating possibility theory into Stackelberg solutions in non-fuzzy environments. This paper shows some theoretical properties and theorems on the formulated fuzzy bilevel programming problems. It is shown that the proposed possibilistic Stackelberg solutions are obtained by using linear or nonlinear programming techniques. The effectiveness of the proposed models is demonstrated by numerical experiments using transportation problems where the upper-level decision maker is a buyer and the lower-level one is a supplier.

A global optimization algorithm for solving the bi-level linear fractional programming problem

Multi-objective linear bi-level multi-follower programming problem (MOBLMFPP) is a special case of two level hierarchical programming problems in which the second level hierarchy includes multiple followers. This paper presents an alternate method based on fuzzy goal programming approach for the solution of multi-objective linear bi-level multi-follower programming (MOLBMFP) problem in which there is no sharing of information among followers. In the proposed FGP model formulation, each of objective functions of each level (leader and follower's level of MOLBMFPP) as well as decision variables at each level are characterised into fuzzy goals. Suitable linear membership functions are defined for each objective function and decision variables. Then minimising the sum of the negative deviational variables of both levels, the highest membership value of each of fuzzy goals is obtained. Existence of compromise optimal solution for MOLBMFPP is established in context of proposed approach. A numerical example is illustrated in support of proposed methodology. A comparative analysis is also carried out in order to show efficiency of proposed method over earlier method for the solution of MOLBMFPP.

BackgroundOptimization procedures to identify gene knockouts for targeted biochemical overproduction have been widely in use in modern metabolic engineering. Flux balance analysis (FBA) framework has provided conceptual simplifications for genome-scale dynamic analysis at steady states. Based on FBA, many current optimization methods for targeted bio-productions have been developed under the maximum cell growth assumption. The optimization problem to derive gene knockout strategies recently has been formulated as a bi-level programming problem in OptKnock for maximum targeted bio-productions with maximum growth rates. However, it has been shown that knockout mutants in fact reach the steady states with the minimization of metabolic adjustment (MOMA) from the corresponding wild-type strains instead of having maximal growth rates after genetic or metabolic intervention. In this work, we propose a new bi-level computational framework--MOMAKnock--which can derive robust knockout strategies under the MOMA flux distribution approximation.MethodsIn this new bi-level optimization framework, we aim to maximize the production of targeted chemicals by identifying candidate knockout genes or reactions under phenotypic constraints approximated by the MOMA assumption. Hence, the targeted chemical production is the primary objective of MOMAKnock while the MOMA assumption is formulated as the inner problem of constraining the knockout metabolic flux to be as close as possible to the steady-state phenotypes of wide-type strains. As this new inner problem becomes a quadratic programming problem, a novel adaptive piecewise linearization algorithm is developed in this paper to obtain the exact optimal solution to this new bi-level integer quadratic programming problem for MOMAKnock.ResultsOur new MOMAKnock model and the adaptive piecewise linearization solution algorithm are tested with a small E. coli core metabolic network and a large-scale iAF1260 E. coli metabolic network. The derived knockout strategies are compared with those from OptKnock. Our preliminary experimental results show that MOMAKnock can provide improved targeted productions with more robust knockout strategies.

The bilevel programming problem (BLPP) is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. To obtain optimality conditions, we reformulate BLPP as a single level mathematical programming problem (SLPP) which involves the value function of the lower level problem. For this mathematical programming problem, it is shown that in general the usual constraint qualifications do not hold and the right constraint qualification is the calmness condition. It is also shown that the linear bilevel programming problem and the minmax problem satisfy the calmness condition automatically. A sufficient condition for the calmness for the bilevel programming problem with quadratic lower level problem and nondegenerate linear complementar¬ity lower level problem are given. First order necessary optimality condition are given using nonsmooth analysis. Second order sufficient optimality conditions are also give...

Optimistic Stackelberg solutions to bilevel linear programming with fuzzy random variable coefficients

Under study is a special class of mixed-integer bi-level programming problem (BLPP) which arises in several areas such as engineering, transportation and control systems. The main characteristic of BLPP is that the outer optimization (upper level) problem is constrained by an inner optimization (lower level) problem. In this paper, a Lagrangean relaxation method is proposed to obtain an acceptable feasible solution by solving a sequence of one-level mixed-integer problems. The computational results on some examples were taken from literatures and randomly generated problems indicate that the method is able to solve the BLPP efficiently.

This paper considers bilevel linear programming with fuzzy random variables. Through the introduction of a novel optimisation model that is to minimise the absolute deviation of degree of possibility and necessity that each fuzzy goal is fulfilled, the original bilevel programming problems with fuzzy random variables can be transformed into deterministic bilevel fractional programming problems. It is shown that the extended Stackelberg solutions can be obtained by solving the transformed bilevel linear programming problem through the combined use of the variable transformation method and the Kth best algorithm. A numerical example of bilevel purchase problem for food retailing is provided to illustrate the usefulness of the proposed methods.

By handling the travel cost function artfully, the authors formulate the transportation mixed network design problem (MNDP) as a mixed-integer, nonlinear bilevel programming problem, in which the lower-level problem, comparing with that of conventional bilevel DNDP models, is not a side constrained user equilibrium assignment problem, but a standard user equilibrium assignment problem. Then, the bilevel programming model for MNDP is reformulated as a continuous version of bilevel programming problem by the continuation method. By virtue of the optimal-value function, the lower-level assignment problem can be expressed as a nonlinear equality constraint. Therefore, the bilevel programming model for MNDP can be transformed into an equivalent single-level optimization problem. By exploring the inherent nature of the MNDP, the optimal-value function for the lower-level equilibrium assignment problem is proved to be continuously differentiable and its functional value and gradient can be obtained efficiently. Thus, a continuously differentiable but still nonconvex optimization formulation of the MNDP is created, and then a locally convergent algorithm is proposed by applying penalty function method. The inner loop of solving the subproblem is mainly to implement an all-or-nothing assignment. Finally, a small-scale transportation network and a large-scale network are presented to verify the proposed model and algorithm.

This paper represents the bi-level linear programming problem in an imprecise and uncertain mixed environment. The aim of this paper is to introduce coefficients of leader and the follower’s objective function and the constraints as the fuzzy random variable. To determine the optimal value of the leader and follower’s objectives a new methodology is developed for bi-level linear programming model in the presence of fuzzy random variable. A numerical example is solved to demonstrate the methodology.

Multilevel optimization problems often arise in various applications (in economics, ecology, power engineering and other areas) when modeling complex systems with a hierarchical structure associated with independent actions of subsystems. The difficulty of analyzing such complex systems requires first of all the study of bilevel models, the management of which would be an integral part of the analysis of more complex systems. In solving bilevel programming problems, an important role is played by the property of partial calmness, the presence of which allows us to reduce the bilevel problem to the classical nonlinear programming problem with a nonsmooth objective function. It is known that linear bilevel programming problems are partially stable. The proof of this property for more complex problems meets difficulties. In particular, our article shows the inaccuracy of some results in this area. The goal of the paper is to obtain some new results in the partial calmness of bilevel programming. In particular, new sufficient conditions for bilevel problems are proved. The results are obtained on the base of Lipschitz-like properties for multivalued mappings. In the paper we propose new sufficient conditions for partial calmness which are based on some modification of the known constraint qualification RCPLD which have been proposed by the researches Andreani, Haeser, Schuverdt and Silva.

In this paper, the triangular fuzzy bilevel programming problem (TFBLPP) is investigated. We transformed TBFLPP to interval fuzzy bilevel linear programming problem by using nearest interval approximation. In the next step, three problem were obtained that by solving them solution TBFLPP obtained.

A novel projected two-binary-variable formulation for unit commitment in power systems

We consider the bilevel programming problem and its optimal value and KKT one level reformulations. The two reformulations are studied in a unified manner and compared in terms of optimal solutions, constraint qualifications and optimality conditions. We also show that any bilevel programming problem where the lower level problem is linear with respect to the lower level variable, is partially calm without any restrictive assumption. Finally, we consider the bilevel demand adjustment problem in transportation, and show how KKT type optimality conditions can be obtained under the partial calmness, using the differential calculus of Mordukhovich.