Practical Bilevel Optimization: Algorithms and ApplicationsThe use of optimization techniques has become integral to the design and analysis of most industrial and socio-economic systems. Great strides have been made recently in the solution of large-scale problems arising in such areas as production planning, airline scheduling, government regulation, and engineering design, to name a few. Analysts have found, however, that standard mathematical programming models are often inadequate in these situations because more than a single objective function and a single decision maker are involved. Multiple objective programming deals with the extension of optimization techniques to account for several objective functions, while game theory deals with the inter-personal dynamics surrounding conflict. Bilevel programming, the focus of this book, is in a narrow sense the combination of the two. It addresses the problern in which two decision makers, each with their individual objectives, act and react in a noncooperative, sequential manner. The actions of one affect the choices and payoffs available to the other but neither player can completely dominate the other in the traditional sense. |
From inside the book
Results 1-5 of 87
Page 13
... corresponding solution * for the lower - level player is any point within the set { ( x1 , x2 ) : x1 + x2 = 1 , x1 ≥ 0 , x2 ≥ 0 } . Thus as in problem ( 1.4a , b ) , the lower - level player is indifferent to a range of points but now ...
... corresponding solution * for the lower - level player is any point within the set { ( x1 , x2 ) : x1 + x2 = 1 , x1 ≥ 0 , x2 ≥ 0 } . Thus as in problem ( 1.4a , b ) , the lower - level player is indifferent to a range of points but now ...
Page 21
... corresponding set of m linear equations in m variables , evaluate the objective function at each feasible combination , and select the best result . For all but the smallest problems , this approach is highly inefficient . For the ...
... corresponding set of m linear equations in m variables , evaluate the objective function at each feasible combination , and select the best result . For all but the smallest problems , this approach is highly inefficient . For the ...
Page 22
... corresponding m × m matrix by B. The matrix B is then nonsingular and can be uniquely determined by solving the ... corresponds to an expression for the vector b as a linear combination of these basis vectors . This interpretation is ...
... corresponding m × m matrix by B. The matrix B is then nonsingular and can be uniquely determined by solving the ... corresponds to an expression for the vector b as a linear combination of these basis vectors . This interpretation is ...
Page 23
... corresponding to whether or not the columns a1 , a2 , ... , ap are linearly independent . Case 1 : Assume a1 , a2 , ap are linearly independent , implying that p ≤ m . If pm , the solution is basic and the proof is complete . If p < m ...
... corresponding to whether or not the columns a1 , a2 , ... , ap are linearly independent . Case 1 : Assume a1 , a2 , ap are linearly independent , implying that p ≤ m . If pm , the solution is basic and the proof is complete . If p < m ...
Page 24
... corresponding component w ; is positive , negative or zero . Because at least one w ; is positive , at least one component will decrease as ɛ is increased . Now , let us increase ɛ to the first point where one or more components become ...
... corresponding component w ; is positive , negative or zero . Because at least one w ; is positive , at least one component will decrease as ɛ is increased . Now , let us increase ɛ to the first point where one or more components become ...
Other editions - View all
Practical Bilevel Optimization: Algorithms and Applications Jonathan F. Bard No preview available - 2010 |
Common terms and phrases
algorithm assumed b₁ backtracking branch and bound c₁x coefficients column complementary slackness computational consider constraints convergence convex convex combination convex functions corresponding cost d₁y defined denote depth-first search direction dual equal equivalent Example exists extreme points fathomed feasible region feasible solution finite follower's problem formulation GABBA given global optimum go to Step implies inducible region inequality infeasible iteration Kuhn-Tucker leader linear BLPP linear program linearly independent live nodes lower bound lower-level problem matrix minimizing mixed-integer nonbasic variables nonlinear programming nonnegative objective function objective function value obtained optimal solution parameter partition pivot primal procedure programming problem Proof Proposition quadratic rational relaxation satisfying search tree Section sequence simplex algorithm simplex method slack variable solve subject to G(x subproblem Table tableau Theorem upper bound vector vertex zero