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Optimisation
Published in Graeme Dandy, David Walker, Trevor Daniell, Robert Warner, Planning and Design of Engineering Systems, 2018
Graeme Dandy, David Walker, Trevor Daniell, Robert Warner
For an LP problem in canonical form the dual problem may be written using the following rules: There is a dual variable corresponding to each primal constraint and a primal variable corresponding to each dual constraint.If the primal problem is of the maximisation type, the dual problem is of the minimisation type and vice versa.All of the constraints in the maximisation problem are of the ‘less than or equal to’ type. All of the constraints in the minimisation problem are of the ‘greater than or equal to’ type.The coefficients of the objective function in the primal problem are the right-hand sides in the dual problem and vice versa.The variables in both problems are non-negative.
A classification recognition method based on an improved kernel function support vector machine
Published in Fei Lei, Qiang Xu, Guangde Zhang, Machinery, Materials Science and Engineering Applications, 2017
Weiwei Xiong, Wei Liang, Mingxia Su
The problem is transformed into a dual problem. That is, under the constraint conditions, Σi=1nyiai=0,ai≥0,i=1,…,n
Optimization
Published in Slobodan P. Simonović, Managing Water Resources, 2012
By reference to the notion of duality, we deepen our understanding of what is really happening in the simplex method. The coefficients of the slack variables in Row 0 of the primal problem at each iteration can be interpreted as trial values of the dual variables. So the simplex method can be seen as an approach that seeks feasibility for the dual problem while maintaining feasibility in the primal problem. As soon as feasible solutions to both problems are obtained, the simplex iterations terminate.
An efficient model for locating solid waste collection sites in urban residential areas
Published in International Journal of Production Research, 2021
Olawale J. Adeleke, M. Montaz Ali
The Lagrangian procedure described in Algorithm 1 calls the subgradient method to solve (19) by initialising with suitable values such that is large enough to move the solution towards the extreme points (Carter and Price 2001). Next, is minimised with fixed values of by solving individual sub-problem and the obtained solutions are then used in the primal problem to find a new set of values for (Balci and Valenzuela 2004). This iterative process continues until a stopping criterion, usually the duality gap, is satisfied. The duality gap measures the difference between the values of the primal problem and the dual problem. At each of the iterations, the values of the multipliers are updated using appropriate techniques. The subgradient optimisation has often been the method of choice among LR researchers in achieving this purpose largely because the generated lower bound is often close enough to the optimal value of the primal problem (Beasley 1993). The method is described as follows: a vector is a sub-gradient of at if .
Towards price-based predictive control of a small-scale electricity network
Published in International Journal of Control, 2020
P. Braun, L. Grüne, C. M. Kellett, S. R. Weller, K. Worthmann
Instead of solving the primal optimisation problem (P) directly, we search for an optimal solution of the dual problem (D) from which an optimal solution of the primal problem is obtained. If the dual function ψ is known, the dual problem (D) can be solved by a gradient ascent method. The key idea is to find a sequence such that ψ(λℓ + 1) > ψ(λℓ) holds for all . Typically, directions satisfying the ascending condition are computed to fulfil this task; e.g. dℓ = ∇ψ(λℓ) is a possible choice. Then, the gradient method is defined by using a sequence of suitable stepsizes. Here, we distinguish between the following two cases: Constant stepsizec > 0: cℓ = c for all .Diminishing stepsize: a sequence satisfying e.g. the harmonic numbers cℓ = 1/(ℓ + 1).
Fenchel–Rockafellar theorem in infinite dimensions via generalized relative interiors
Published in Optimization, 2023
D. V. Cuong, B. S. Mordukhovich, N. M. Nam, G. Sandine
Duality theory has a central role in optimization theory and its applications. From a primal optimization problem with the objective function defined in a primal space, a dual problem in the dual space is formulated with the hope that the new problem is easier to solve, while having a close relationship with the primal problem. Its important role in optimization has made duality theory attractive for extensive research over the past few decades; see, e.g. [1–19] and the references therein.