Lagrangian relaxation

About: Lagrangian relaxation is a research topic. Over the lifetime, 4244 publications have been published within this topic receiving 124805 citations.

Short-term generation scheduling with transmission and environmental constraints using an augmented Lagrangian relaxation

Abstract: This paper proposes a new approach based on augmented Lagrangian relaxation for short term generation scheduling problems with transmission and environmental constraints. In this method, the power system constraints, e.g. load demand, spinning reserve, transmission capacity and environmental constraints, are relaxed by using Lagrangian multipliers, and quadratic penalty terms associated with power system load demand balance are added to the Lagrangian objective function. Then, the decomposition and coordination technique is used, and nonseparable quadratic penalty terms are replaced by linearization around the solution obtained from the previous iteration. In order to improve the convergence property, the exactly convex quadratic terms of decision variables are added to the objective function as strongly convex, differentiable and separable auxiliary functions. The overall problem is decomposed into N subproblems, multipliers and penalty coefficients are updated in the dual problem and power system constraints are satisfied iteratively. The corresponding unit commitment subproblems are solved by dynamic programming, and the economic dispatch with transmission and environmental constraints is solved by an efficient network flow programming algorithm. The augmented Lagrangian relaxation method enhanced by the decomposition and coordination techniques avoids oscillations associated with piece-wise linear cost functions. Numerical results indicate that the proposed approach is fast and efficient in dealing with numerous power system constraints. >

Towards a more rigorous and practical unit commitment by Lagrangian relaxation

Abstract: A Lagrangian relaxation algorithm for power system generator unit commitment is proposed. The algorithm proceeds in three phases. In the first phase, the Lagrangian dual of the unit commitment is maximized by standard subgradient techniques. The second phase finds a reserve-feasible dual solution, followed by a third phase of economic dispatch. A mathematically based, systematic and generally applicable procedure to search for a reserve-feasible dual solution is presented. The algorithm has been tested on systems of up to 100 units to be scheduled over 168 hours, giving reliable performance and low execution times. Both spinning and time-limited reserve constraints are treated. >

The Dynamic Berth Allocation Problem for a Container Port

Abstract: This paper addresses the problem of determining a dynamic berth assignment to ships in the public berth system. While the public berth system may not be suitable for most container ports in major countries, it is desired for higher cost-effectiveness in Japan's ports. The berth allocation to calling ships is a key factor for efficient public berthing. However, it is not calculated in polynomially-bounded time. To obtain a good solution with considerably small computational effort, we developed a heuristic procedure based on the Lagrangian relaxation of the original problem. We conducted a large amount of computational experiments which showed that the proposed algorithm is adaptable to real world applications.

An Exact Two-Dimensional Non-Guillotine Cutting Tree Search Procedure

Abstract: We consider the two-dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut. We develop a Lagrangean relaxation of a zero-one integer programming formulation of the problem and use it as a bound in a tree search procedure. Subgradient optimization is used to optimize the bound derived from the Lagrangean relaxation. Problem reduction tests derived from both the original problem and the Lagrangean relaxation are given. Incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved.

The dynamic berth allocation problem for a container port

Abstract: This paper addresses the problem of determining a dynamic berth assignment to ships in the public berth system. While the public berth system may not be suitable for most container ports in major countries, it is desired for higher cost-effectiveness in Japan's ports. The berth allocation to calling ships is a key factor for efficient public berthing. However, it is not calculated in polynomially-bounded time. To obtain a good solution with considerably small computational effort, we developed a heuristic procedure based on the Lagrangian relaxation of the original problem. We conducted a large amount of computational experiments which showed that the proposed algorithm is adaptable to real world applications.