Diffusion sampling schemes: A generalized methodology with nongeometric criteria.

This work deals with a class of nonlinear bilevel programming problems with nonconvex objective functions, in which the follower objective is a function of linear expression of all variables and the follower constraints are linear. For the leader functions, there are no any restrictions on the convexity as well as the differentiability. The distinguished feature of the problem is the nonconvexity of the follower objective function, which breaks through the barrier that the follower must be convex or concave in literature. First, for any fixed leader variable x, two linear programming are got from the follower and used to obtain the follower optimal solution y. In addition, in order to avoid solving directly the follower problem for each x, a real-binary encoding scheme is given which consists of x and the bases of two linear programming. Finally, a new crossover operator is designed based on the characteristics of the mixed encoding, and a novel genetic algorithm is proposed. The numerical results on 15 examples illustrate that the proposed algorithm is effective and stable.

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.

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.

Smoothing inertial projection neural network for minimization [formula omitted] in sparse signal reconstruction

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.

The problem of sparse signal recovery from lower number of observations is formalized as a constrained minimization problem. Due to the dissimilar requirements of signal processing, the constraints and objective functions differ from each other. This paper presents an algorithm scheme base on alternating iteration to deal with the nonconvex objective function. The proposed method employs the log-penalty and proximity operator to seek the sparse solution. In each iteration, the introduction of nonconvex penalty decreases the penalization of large coefficients which contributes to a faster decrease of the function value. By making use of proximity operator, it reduces the computational complexity and achieves less number of iterations. In addition, the experimental results illustrate the effectiveness of the proposed method as well as the drawbacks of the algorithm.

Adjoint-based well placement optimisation for Enhanced Oil Recovery (EOR) under geological uncertainty: From seismic to production

Differential privacy has been a common framework that provides an effective method of establishing privacy-guaranteed machine learning. Extensive research work has focused on differential privacy stochastic gradient descent (SGD-DP) and its variants under distributed machine learning to improve training efficiency and protect privacy. However, SGD-DP relies on the premise of convex optimization. In large-scale distributed machine learning, the objective function may be more a non-convex objective function, which not only makes the gradient calculation difficult and easy to fall into local optimization. It’s difficult to achieve truly global optimization. To address this issue, we propose a novel differential privacy optimization algorithm based on quantum particle swarm theory that suitable for both convex optimization and non-convex optimization. We further comprehensively apply adaptive contraction–expansion and chaotic search to overcome the premature problem, and provide theoretical analysis in terms of convergence and privacy protection. Also, we verify through experiments that the actual application performance of the algorithm is consistent with the theoretical analysis.

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.

A new version of accelerated gradient descent is proposed. The method does not require any a priori information on the objective function, uses a linesearch procedure for convergence acceleration in practice, converge according to well-known lower bounds for both convex and nonconvex objective functions, and has primal-dual properties. A universal version of this method is also described.

A Surrogate-Based Optimization Method with Dynamic Adaptation for High-Dimensional Mixed-Integer Problems

We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed $$l_1$$ (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates $$l_0$$ and $$l_1$$ norms through a nonnegative parameter $$a \in (0,+\infty )$$ , similar to $$l_p$$ with $$p \in (0,1]$$ , and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem, and discuss the exact recovery of $$l_0$$ norm minimal solution based on the null space property (NSP). We then prove the stable recovery of $$l_0$$ norm minimal solution if the sensing matrix A satisfies a restricted isometry property (RIP). We formulated a normalized problem to overcome the lack of scaling property of the TL1 penalty function. For a general sensing matrix A, we show that the support set of a local minimizer corresponds to linearly independent columns of A. Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized constrained and unconstrained problems in CS. The DCATL1 algorithm involves outer and inner loops of iterations, one time matrix inversion, repeated shrinkage operations and matrix-vector multiplications. The inner loop concerns an $$l_1$$ minimization problem on which we employ the Alternating Direction Method of Multipliers. For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value $$a=1$$ , and compare DCATL1 with other CS algorithms on two classes of sensing matrices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). Among existing algorithms, the iterated reweighted least squares method based on $$l_{1/2}$$ norm is the best in sparse recovery for Gaussian matrices, and the DCA algorithm based on $$l_1$$ minus $$l_2$$ penalty is the best for over-sampled DCT matrices. We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with $$l_1$$ minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix A. DCATL1 is also competitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.

The algorithms based on the technique of optimal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-thresholding (OT) were recently proposed for signal recovery, and they are very different from the traditional family of hard thresholding methods. However, the computational cost for OT-based algorithms remains high at the current stage of their development. This stimulates the development of the so-called natural thresholding (NT) algorithm and its variants in this paper. The family of NT algorithms is developed through the first-order approximation of the so-called regularized optimal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-thresholding model, and thus the computational cost for this family of algorithms is significantly lower than that of the OT-based algorithms. The guaranteed performance of NT-type algorithms for signal recovery from noisy measurements is shown under the restricted isometry property and concavity of the objective function of regularized optimal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-thresholding model. Empirical results indicate that the NT-type algorithms are robust and very comparable to several mainstream algorithms for sparse signal recovery.

Background: The idea behind the Internet of Things is to bring the virtual world into the physical one by connecting commonplace items. With the help of the Internet of Things (IoT), it is possible to remotely sense or control objects through preexisting network infrastructure. This opens up possibilities for computer-based systems to integrate with the physical world, which in turn improves efficiency, accuracy, and economic benefit while reducing the need for human intervention. Objective: The purpose of this patent study is to investigate how a (NSGA-II) multi-objective genetic algorithm might be utilized to optimize the execution of an Internet of Things (IoT) temperature monitoring Box-Type Solar Cooker (BTSC). To determine the best set of output parameters for an IoT temperature monitoring box-type solar cooker, (NSGA-II) multi-objective genetic algorithms are used to perform optimizations of the figure of merits (F2), cooking power, cooker efficiency, and final water temperature. Methods: The present research work involves the development of a Wi-Fi module system integrated with a smart temperature monitoring system for a BTSC. Keeping track of the temperature data from different locations in the BTSC through the IoT system was the primary objective of this project. A waterproof temperature sensor (DS18B20) was used to keep monitoring. After that, the data was shown on an LCD, stored on a microSD card, and made available through a smartphone. The Blynk Applications' IoT was employed. Using existing data, regression-based computational models are developed to describe the complex correlations between the decision-processing parameters and the input parameters of an IOT-based solar cooker. These models are applied in the objective functions after determining that a genetic algorithm is more appropriate for the problem. To forecast the optimal values about the figure of merits (F2), cooking power, cooker efficiency, and final water temperature, the Pareto fronts have been developed. Results: We compare the values of response variables that were gathered experimentally with the values that were predicted by NSGA-II. The predicted values are found to be quite close to experimental values. This indicates that the multi-objective optimization method, as used in this study, has very good prediction performance. The test results are graphically shown using the error bar. Therefore, it is clear that the optimization process used to adjust the parameters of the solar cooker's performance has been quite effective. According to the findings of the experiment, the temperature at which a cooking pot remained stagnant on average was 158°C. It was determined that the cooker was of class A based on the values of the first figure of merit (F1), the second figure of merit (F2), and the cooking power (P), which were respectively 0.132, 0.359, and 86.108 W. Therefore, the thermal efficiency of the IoT-base temperature monitoring box type solar cooker is 39.99 %. Conclusion: The findings of this inquiry furthermore produced the outcome that the model provided can be applied conveniently with a confidence level of 95% to calculate the figure of merits (F2), cooking power, cooker efficiency, and final water temperature value of an Internet of Things-based temperature monitoring BTSC. The performance of IoT-based BTSC is optimized by providing real- time monitoring and data visualization, ultimately improving their efficiency and reliability. This research provides an educational tool to promote awareness and understanding of renewable energy sources and their potential benefits.

In this paper, the highly nonlinear planetary-entry optimal control problem is formulated as a sequence of convex problems to facilitate rapid solution. The nonconvex control constraint is avoided by introducing a new state variable to the original three-dimensional equations of motion. The nonconvex objective function and path constraints are convexified by first-order Taylor-series expansions, and the nonconvex terms in the dynamics are approximated by successive linearizations. A successive solution procedure is developed to find an approximated solution to the original problem, and its convergence is discussed. In each iteration, a convex optimization problem is solved by the state-of-the-art interior-point method with deterministic convergence properties. Finally, the proposed method is verified and compared to a general-purpose optimal control solver by numerical solutions of minimum terminal-velocity and minimum heat-load entry problems. The sequential method converges to accurate solutions with faster speed than the general-purpose solver using MATLAB on a desktop computer with a 64-bit operating system and an Intel Xeon E3-1225 V2 3.2 GHz processor, which demonstrates its potential real-time application for computational guidance.