China
Africa
Explore chapters and articles related to this topic
Exploiting the Flexibility Value of Virtual Power Plants through Market Participation in Smart Energy Communities
Published in Ehsan Heydarian-Forushani, Hassan Haes Alhelou, Seifeddine Ben Elghali, Virtual Power Plant Solution for Future Smart Energy Communities, 2023
Georgios Skaltsis, Stylianos Zikos, Elpiniki Makri, Christos Timplalexis, Dimosthenis Ioannidis, Dimitrios Tzovaras
Many studies model the mathematical problem as a mixed-integer linear problem, through the fast execution time that it is provided. However, some authors formulate the VPP issue with nonlinear constraints, and due to nonlinearity, they apply several techniques trying to avoid the possibility of generating local optimal solutions. Regarding discrete optimization problems, a widely used algorithm is the branch-and-bound technique [16]. The branch-and-bound method recursively divides the search space into smaller spaces, called branches by using estimated bounds to limit the number of possible solutions. Nevertheless, the formed tree may become enormous; in that case, the available memory will be drained. Dynamic programming is another optimization method that breaks down a complex problem into a group of simpler sub-problems, solving them separately just once and then storing their solution. Therefore, the optimal solution will be a combination of sub-problems' outcomes.
Partial Digest Problem
Published in Adwitiya Sinha, Megha Rathi, Smart Healthcare Systems, 2019
Urvi Agarwal, Sanchi Prakash, Harshit Agarwal, Prantik Biswas, Suma Dawn, Aparajita Nanda
Branch and bound refers to a class of algorithm that is used for optimally solving combinatorial problems. This algorithm does the same by forming a set X and adding elements to it if they satisfy certain conditions. As seen in both of the earlier exhaustive searches, every element selected is independent of the previously added elements. This algorithm focuses on examining the previously added elements before adding a new element to the solution set. Initially, a set X with only element 0 is declared. We proceed by adding the width of set L to X and remove it from L. Taking the present highest element from set L, check whether its pairwise distances with elements of X exist in X. If they do, then the particular element is added to set X and the algorithm proceeds by repeating the steps until L is empty. If they don’t, then the next highest element is selected and the earlier steps are repeated.
Optimization
Published in Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer, Classical and Modern Numerical Analysis, 2009
Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer
The basic process in branch and bound algorithms is to bound the optimum above, bound the ranges of the objective and constraints over subregions, then reject those subregions over which the range bounds show that the problem either must be infeasible or the objective function must be greater than the upper bound on the optimum. Although practical for some problems, this simple basic technique requires much computation, especially in higher dimensions, where, with n variables, bisecting uniformly so the resulting box has half the diameter of the original box results in 2n sub-boxes. For this reason, various other techniques are used in practical software systems to reduce the volume of region to be searched. We now mention a few of these.
Non-convex regularization and accelerated gradient algorithm for sparse portfolio selection
Published in Optimization Methods and Software, 2023
Qian Li, Wei Zhang, Guoqiang Wang, Yanqin Bai
To obtain an ideal portfolio which performs well out-of-sample, while less sensitive to the inputs and cheap to maintain and monitor, sparsity plays an important role in the formulation of portfolios. Naturally, the cardinality constraint is introduced to limit the total number of nonzero positions, where . The cardinality constrained portfolio selection model is developed as follows: The main approach for solving CCMV is to first reformulate it into a mixed integer programming and then use the branch-and-bound algorithm [10]. As we know, the branch-and-bound algorithm suffers from expensive computational cost for solving large-scale problems.
More MILP models for hybrid flow shop scheduling problem and its extended problems
Published in International Journal of Production Research, 2020
Leilei Meng, Chaoyong Zhang, Xinyu Shao, Biao Zhang, Yaping Ren, Wenwen Lin
Mathematical model is the basis of understanding scheduling problems. It can describe the objectives, constraints and all the other characteristics of the scheduling problem clearly and serves as the key to mine scheduling knowledge and heuristic rules (Unlu and Mason 2010). Mathematical models can be solved with branch and bound method, dynamic search, and branch-and-cut among others (Naderi, Gohari, and Yazdani 2014). The branch-and-cut method is implemented in CPLEX of this paper and is used to solve MILP problems. The branch-and-cut method is the combination of cutting plane and branch-and-bound methods, which works by running a branch-and-bound algorithm and using cutting planes to tighten the linear programming relaxations (Meng et al. 2019b). The efficiency of the mathematical model depends on many factors such as the number of decision variables, the dimensionality of decision variables, the number of constraints and constraints’ tightness. As efficient mathematical models can solve problems fast and get better results within the same time, the construction of such models is of great significance.
Permutation flowshop scheduling with simple linear deterioration
Published in Engineering Optimization, 2019
Lin-Hui Sun, Chen-Chen Ge, Wei Zhang, Ji-Bo Wang, Yuan-Yuan Lu
In order to solve the problem optimally, a branch-and-bound algorithm can be proposed. For the branch-and-bound algorithm, the initialization solution (upper bound) can be obtained by the SLDR (WSLDR) rule for and (), and a depth-first search is adopted in the branching procedure. The efficiency of the branch-and-bound algorithm depends largely on the effectiveness of calculating lower bounds for partial solutions. Let be a schedule, where is the scheduled part and is the set of so far unscheduled jobs. It is assumed that there are h jobs in , hence the th job on machine is Similarly, where