A real-binary coded genetic algorithm for solving nonlinear bilevel programming with nonconvex objective functions

In this paper, a special nonlinear bilevel programming problem (nonlinear BLPP) is transformed into an equivalent single objective nonlinear programming problem. To solve the equivalent problem effectively, we first construct a specific optimization problem with two objectives. By solving the specific problem, we can decrease the leader's objective value, identify the quality of any feasible solution from infeasible solutions and the quality of two feasible solutions for the equivalent single objective optimization problem, force the infeasible solutions moving toward the feasible region, and improve the feasible solutions gradually. We then propose a new constraint-handling scheme and a specific-design crossover operator. The new constraint-handling scheme can make the individuals satisfy all linear constraints exactly and the nonlinear constraints approximately. The crossover operator can generate high quality potential offspring. Based on the constraint-handling scheme and the crossover operator, we propose a new evolutionary algorithm and prove its global convergence. A distinguishing feature of the algorithm is that it can be used to handle nonlinear BLPPs with nondifferentiable leader's objective functions. Finally, simulations on 31 benchmark problems, 12 of which have nondifferentiable leader's objective functions, are made and the results demonstrate the effectiveness of the proposed algorithm.

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.

An augmented Lagrangian multiplier method based on a CHKS smoothing function for solving nonlinear bilevel programming problems

Mathematical modelling of cellular metabolism plays an important role in understanding biological functions and providing identification of targets for biotechnological modification. This paper proposes a nonlinear bilevel programming (NBP) model to infer the objective function of anaerobic glycerol metabolism in Klebsiella Pneumoniae (K. Pneumoniae) for 1, 3-propanediol (1, 3-PD) production. Based on the Kuhn-Tucker optimality condition of the lower level problem, NBP is transformed into a nonlinear programming with complementary and slackness conditions. The authors give the existence theorem of solutions to NBP. An efficient algorithm is proposed to solve NBP and its convergence is also simply analyzed. Numerical results reveal some interesting conclusions, e.g., biomass production is the main force to drive glycerol metabolism, and the objective functions, which are obtained in term of several different groups of flux distributions, are similar.

Profit Allocation in the Global Supply Chain with Transfer Pricing and Exchange Rate

Economic dispatch is an important non-linear optimization task in power systems. In this process, the total power demand is distributed amongst the generating units such that each unit satisfies its generation limit constraints and the cost of power production is minimized. This paper presents an over view of three optimization algorithms namely real coded genetic algorithm, particle swarm optimization and a relatively new optimization technique called bat algorithm. This study will further propose modifications to the original bat. Simulations are carried out for two test cases. First is a six-generator power system with a simplified convex objective function. The second test case is a five-generator system with a non-convex objective function. Finally the results of the modified algorithm are compared with the results of genetic algorithm, particle swarm and the original bat algorithm. The results demonstrate the improvement in the Bat Algorithm.

In this paper we propose an iterative algorithm for solving the problem of extracting a sparse source signal when a reference signal for the desired source signal is available. In the proposed algorithm, a nonconvex objective function is used for measuring the diversity (antisparsity) of the desired source signal. The nonconvex function is locally replaced by a quadratic convex function. This results in a simple iterative algorithm. The proposed algorithm has two different versions, depending on the measure of closeness between the extracted source signal and the reference signal. The proposed algorithm has useful applications to EEG/MEG signal processing. This is demonstrated by an example in which eye blink artifacts are automatically removed from a real EEG data.

Unmixing of hyperspectral data is an area of major research because the information it provides is utilized in plethora of fields. The year of 2006 witnessed the emergence of Compressed Sensing algorithm which was later used to spearhead research in umixing problems. Later, the notion of \(\ell _p\) norms \(0< p < 1\) and other non-smooth and non-convex penalty function were used in place of the traditional convex \(\ell _1\) penalty. Dealing with optimization problems with non-convex objective function is rather difficult as most methodologies often get stuck at local optima. In this paper, a parameterised non-convex penalty function is used to induce sparsity in the unknown.The parameters of penalty function can be adjusted so as to make the objective function convex, thus resulting in the possibility of finding a global optimal solution. Here ADMM algorithm is utilized to arrive at the final iterative algorithm for the unmixing problem. The algorithm is tested on synthetic data set, generated from the spectral library provided by US geological survey. Different parametric penalty functions like \(\log \) and \(\arctan \) are used in the algorithm and is compared with the traditional \(\ell _1\) penalties, in terms of the performance measures RSNR and PoS. It was observed that the non-convex penalty functions out-performs the \(\ell _1\) penalty in terms of the aforementioned measures.

Budget constraints arise in many computer vision problems. Computational costs limit many automated recognition systems while crowdsourced systems are hindered by monetary costs. We leverage wide variability in image complexity and learn adaptive model selection policies. Our learnt policy maximizes performance under average budget constraints by selecting “cheap” models for low complexity instances and utilizing descriptive models only for complex ones. During training, we assume access to a set of models that utilize features of different costs and types. We consider a binary tree architecture where each leaf corresponds to a different model. Internal decision nodes adaptively guide model-selection process along paths on a tree. The learning problem can be posed as an empirical risk minimization over training data with a non-convex objective function. Using hinge loss surrogates we show that adaptive model selection reduces to a linear program thus realizing substantial computational efficiencies and guaranteed convergence properties.Keywordstest-time budgetadaptive model selectioncost-sensitive learning

A neurodynamic optimization technique based on overestimator and underestimator functions for solving a class of non-convex optimization problems

Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect

Particle swarm optimization (PSO) algorithm has been developing rapidly and has been applied widely since it was introduced, as it is easily understand and realized. For nonlinear bilevel programming whose leader is a nonlinear function, a hybrid PSO algorithm with a simplex algorithm is presented. So far using PSO to solve the nonlinear bilevel programming problem has not been found in the literature. Some numerical examples are given to verify the effectiveness of proposed approach.

A neural network approach for solving nonlinear bilevel programming problem

The aim of this paper is to show that geometrical criteria for designing multishell -space sampling procedures do not necessarily translate into reconstruction matrices with high figures of merit commonly used in the compressed sensing theory. In addition, we show that a well-known method for visiting k-space in radial three-dimensional acquisitions, namely, the Spiral Phyllotaxis, is a competitive initialization for the optimization of our nonconvex objective function. We propose the gradient design method WISH (WeIghting SHells) which uses an objective function that accounts for weighted distances between gradients within M-tuples of consecutive shells, with ranging between 1 and the maximum number of shells . All the -tuples share the same weight . The objective function is optimized for a sample of these weights, using Spiral Phyllotaxis as initialization. State-of-the-art General Electrostatic Energy Minimization (GEEM) and Spherical Codes (SC) were used for comparison. For the three methods, reconstruction matrices of the attenuation signal using MAP-MRI were tested using figures of merit borrowed from the Compressed Sensing theory (namely, Restricted Isometry Property -RIP- and Coherence); we also tested the gradient design using a geometric criterion based on Voronoi cells. For RIP and Coherence, WISH got better results in at least one combination of weights, whilst the criterion based on Voronoi cells showed an unrelated pattern. The versatility provided by WISH is supported by better results. Optimization in the weight parameter space is likely to provide additional improvements. For a practical design with an intermediate number of gradients, our results recommend to carry out the methodology here used to determine the appropriate gradient table.

This paper considers planar location problems with rectilinear distance and barriers where the objective function is any convex, nondecreasing function of distance. Such problems have a non-convex feasible region and a nonconvex objective function. Based on an equivalent problem with modified barriers, derived in a companion paper [3], the non convex feasible set is partitioned into a network and rectangular cells. The rectangular cells are further partitioned into a polynomial number of convex subcells, called convex domains, on which the distance function, and hence the objective function, is convex. Then the problem is solved over the network and convex domains for an optimal solution. Bounds are given that reduce the number of convex domains to be examined. The number of convex domains is bounded above by a polynomial in the size of the problem.