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Pseudo-inverse learners (PILs) are a kind of feedforward neural network trained with the pseudoinverse learning algorithm, which can be traced back to 1995 originally. PIL is an approach for nongradient descent learning, and its main advantage is the lower computational cost and fast learning procedure, which is especially relevant in the edge computing research field. However, PIL is mostly applied to a deterministic learning problem, while in the real world, the greatest case that is of concern is the uncertainty learning problem. In this work, under the framework of the synergetic learning system (SLS), we introduce an approximated synergetic learning scheme, which can transform uncertainty learning into deterministic learning. We call this new learning framework the Bayesian PIL, and the advantages are also demonstrated in this work.

A new three-dimensional missile guidance law to control the impact vector against a stationary target is proposed. The composite guidance law has two well-known components: Apollo descent guidance and trajectory shaping guidance. These respectively linear and planar guidance laws are combined to achieve a specified impact direction. The main idea is to define an impact plane and to steer the missile onto this plane using Apollo descent guidance while concurrently performing trajectory shaping with reference to the impact plane. The resulting guidance law is expressed by a single equation in vector form, which is straightforward to implement. Because it originates from an optimal formulation, the performance of the proposed technique is expected to be satisfactory. This is confirmed by comparative simulation runs, which also involve the method known as generalized explicit guidance.

A gradient-based search method (GBSM) is developed to solve multi-objective optimization problems. It uses the multi-objective gradient information to construct descent directions, i.e., Pareto descent directions (PDDs), to accelerate the convergence. In addition, a multi-objective evolutionary algorithm based on decomposition is adopted to improve the diversity. The comparisons between GBSM with several selected multi-objective evolutionary algorithms and gradient based algorithms on benchmark functions indicate that the proposed method performs competitively and effectively.

Clustering is a common task in data mining for constructing well-separated groups (clusters) from a large set of data points. The balanced minimum sum-of-squares clustering problem is a variant of the classic minimum sum-of-squares clustering (MSSC) problem and arises from broad real-life applications where the cardinalities of any two clusters differ by at most one. This study presents the first memetic algorithm for solving the balanced MSSC problem. The proposed algorithm combines a backbone-based crossover operator for generating offspring solutions and a responsive threshold search that alternates between a threshold-based exploration procedure and a descent-based improvement procedure for improving new offspring solutions. Numerical results on 16 real-life datasets show that the proposed algorithm competes very favorably with several state-of-the-art methods from the literature. Key components of the proposed algorithm are investigated to understand their effects on the performance of the algorithm.

This study aims to discuss the steps of Maudu'i interpretation of the Qur'an. This method has steps in interpreting the Qur'an, and usually responds to the challenges of the times, is more practical, systematic, dynamic and easy to understand in its entirety by the general public. This method is relatively new and is considered actual in interpreting the Qur'an departing from a logical unity and interrelated with one another. The method used in this research is descriptive, a research method that seeks to analyze and describe something, such as existing conditions or relationships, developing opinions, and or ongoing processes. Based on this method, in this study, the author will attempt to reveal experts' understanding regarding the steps of maudu'i interpretation. From this theory, it can be seen that to get a complete understanding of the Qur'an according to a certain theme, we must take certain steps, starting from choosing or determining the topic of the Qur'an in a maudhu`i manner, collecting verses that relating to this problem, coherently compiling the verses according to the chronology of their descent, knowing the correlation (munasabah) of these verses in each letter, arranging the theme of discussion in an appropriate framework, completing the discussion and description with the hadith, and study the verses thematically and thoroughly.

We consider relative or subjective optimization problems where the goal function and feasible set are dependent on the current state of the system under consideration. In general, they are formulated as quasi-equilibrium problems, hence finding their solutions may be rather difficult. We describe a rather general class of relative optimization problems in metric spaces, which in addition depend on the starting state. We also utilize quasi-equilibrium type formulations of these problems and show that they admit rather simple descent solution methods. This approach gives suitable trajectories tending to a relatively optimal state. We describe several examples of applications of these problems. Preliminary results of computational experiments confirmed efficiency of the proposed method.

In this paper, we present a new hybrid conjugate gradient (CG) approach for solving unconstrained optimization problem. The search direction is a hybrid form of the Fletcher-Reeves (FR) and the Dai-Yuan (DY) CG parameters and is close to the direction of the memoryless Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton approach. Independent of the line search, the search direction of the new approach satisfies the descent condition and possess the trust region. We establish the global convergence of the approach for general functions under the Wolfe-type and Armijo-type line search. Using the CUTEr library, numerical results show that the propose approach is more efficient than some existing approaches. Furthermore, we give a practical application of the new approach in optimizing risk in portfolio selection.

This research is focused on Unconstrained Optimization problems. Among a number of methods that can be used to solve Unconstrained Optimization problems we have worked on Gradient and Coordinate Descent methods. Step size plays an important role for optimization. Here we have performed numerical experiment with Gradient and Coordinate Descent method for several step size choices. Comparison between different variants of Gradient and Coordinate Descent methods and their efficiency are demonstrated by implementing in loss functions minimization problem.

This article presents a new and fast indirect method for solving fuel-optimal descent trajectories in the gravitational field of an irregularly-shaped asteroid. The costates associated with a two-impulse descent trajectory are used as approximate costates for the costate initialization of a finite-thrust fuel-optimal trajectory planning problem. The approximate solutions of the initial velocity and mass costates are given in analytical forms. A simple shooting equation is derived to solve the approximate position costates. The two-impulsive descent trajectory is solved by an irregular gravitational Lambert solver. Based on the Lambert solver and approximate costates, an algorithm is proposed for trajectory optimization with varying time of flight. The effectiveness and efficiency of the proposed method are tested through the simulation of landing on the asteroid 433 Eros. Additionally, a computationally efficient approximate model is employed to replace the polyhedral model, which greatly improves the computational speed and enhance the ability to address uncertainties.

In this study, we focus mainly on the problem of finding the minimum-length path through a set of circular regions by a fixed-wing unmanned aerial vehicle. Such a problem is referred to as the Dubins traveling salesman problem with neighborhoods (DTSPN). Algorithms developed in the literature for solving DTSPN either are computationally demanding or generate low-quality solutions. To achieve a better trade-off between solution quality and computational cost, an efficient gradient-free descent method is designed. The core idea of the descent method is to decompose DTSPN into a series of subproblems, each of which consists of finding the minimum-length path of a Dubins vehicle from a configuration to another configuration via an intermediate circular region. By analyzing the geometric properties of the subproblems, we use a bisection method to solve the subproblems. As a result, the descent method can efficiently address DTSPN by successively solving a series of subproblems. Finally, several numerical experiments are carried out to demonstrate the descent method in comparison with several existing algorithms.

In this paper, we suggest a descent modification of the conjugate gradient method which converges globally provided that the exact minimization condition is satisfied. Preliminary numerical experiments on some benchmark problems show that the method is efficient and promising.

We propose a global optimization algorithm hybridizing a version of Bayesian global search with local minimization. The implementation of Bayesian algorithm is based on the simplician partition of the feasible region. Our implementation is free from the typical computational complexity of the standard implementations of Bayesian algorithms. The local minimization counterpart improves the efficiency of search in the indicated potential basins of global minimum. The performance of the proposed algorithm is illustrated by the results of a numerical experiment.

NN-Descent is a classic k-NN graph construction approach. It is still widely employed in machine learning, computer vision, and information retrieval tasks due to its efficiency and genericness. However, the current design only works well on CPU. In this paper, NN-Descent has been redesigned to adapt to the GPU architecture. A new graph update strategy called selective update is proposed. It reduces the data exchange between GPU cores and GPU global memory significantly, which is the processing bottleneck under GPU computation architecture. This redesign leads to full exploitation of the parallelism of the GPU hardware. In the meantime, the genericness, as well as the simplicity of NN-Descent, are well-preserved. Moreover, a procedure that allows to k-NN graph to be merged efficiently on GPU is proposed. It makes the construction of high-quality k-NN graphs for out-of-GPU-memory datasets tractable. Our approach is 100-250× faster than the single-thread NN-Descent and is 2.5-5× faster than the existing GPU-based approaches as we tested on million as well as billion scale datasets.

Spectral conjugate gradient methods are considered as an efficient family of conjugate gradient methods to solve unconstrained optimization problems. In this work, based on the scaled memoryless BFGS update, a spectral Polak–Ribiere–Polyak conjugate gradient method is proposed. It is established that the method ensures the sufficient descent property independent of line search techniques as well as convexity of the objective function. The global convergence of the method is shown with the strong Wolfe line search strategy for general functions. Finally, the practical competencies of the method are investigated by numerical experiments on a set of test functions of CUTEr library. The results demonstrate computational efficiency of the proposed method.

Momentum method has been used extensively in optimizers for deep learning. Recent studies show that distributed training through K-step averaging has many nice properties. We propose a momentum method for such model averaging approaches. At each individual learner level traditional stochastic gradient is applied. At the meta-level (global learner level), one momentum term is applied and we call it block momentum. We analyze the convergence and scaling properties of such momentum methods. Our experimental results show that block momentum not only accelerates training, but also achieves better results.

Objective to implement the neighborhood descent heuristic approach into solving the lotsizing problem. Methods a variable neighborhood descent heuristic approach and a framework of lotsizing problem. Results dynamic lotsizing problem is supposed to be an NPhard problem ndash a problem for which there is no known polynomial algorithm so the time to find a solution grows with a problem size. In the article we explain thenbsp basic lotsizing lotsizing in closedloop describe methods used by other researchers and provide reasoning for using a variable neighborhood descent heuristic method. Implementation of the method is presented in a numerical example. Application of the method significantly reduces estimation time when searching for lotsizing problem solution. Application of the method significantly reduces estimation time when searching for lotsizing problem solution. Scientific novelty the method may significantly reduce time expenses during the solution of the lotsizing problem. Practical significance the method may be implemented for solving the lotsizing problem in various industries such as pulp and paper consumer goods industry and heavy industry.

The notion and the theory of constructing planetary powered descent guidance laws based on fractional polynomials are introduced. Along the way a new perspective to the classical Apollo powered descent guidance designs is gained. As a foundation of this work, a theory is first developed on how to derive an explicit powered descent guidance law by selecting the feedback gains in a unique way in an implicit tracking guidance law. Then by employing a particular profile of the reference thrust acceleration in the tracking law with a special set of gains, a large family of powered descent guidance laws is obtained that offer the flexibility in trajectory shaping and thrust characteristics by two adjustable parameters. An equivalence theory is established to show that each in this family of guidance laws is actually the explicit guidance law from a fractional-polynomial thrust acceleration profile. This insight provides a perfect explanation to the predictable behavior and good targeting performance of these guidance laws. The well-known Apollo lunar descent guidance and E-guidance laws are revealed to be two members of this much broader class of fractional-polynomial powered descent guidance laws.

This study aims to analyze the k-value on K nearest neighbor classification. k-value is the distance used to find the closest data to label the class from the testing data. Each k-value can produce a different class label against the same testing data. The variants of k-value that we use are k=3, k=5 and k=7 to find the best k-value. There are 2 classes that are used in this research. Both classes are stairs descent and floor classes. The gray level co-occurrence matrix method is used to extract features. The data we use comes from videos obtained from the camera on the smart wheelchair taken by the frame. Refer to the results of our tests, the best k-value is obtained when using k=7 and angle 0° with accuracy is 92.5%. The stairs descent detection system will be implemented in a smart wheelchair

A variation of the One-to-one Pickup and Delivery Problem (OPDP) in connected graphs, the Split Demand One-to-one Pickup and Delivery Problem with the Shortest-path Transport along Real-life Paths (SDOPDPSTRP) is abstracted from passenger train operation plans based on networks. Unlike the classical OPDP, in the SDOPDPSTRP: the demands can be split and must be transported along the shortest path according to passengers requirements and vehicles should travel along a real-life path. A new kind of integer programming model is formulated for the SDOPDPSTRP based on the connection relationship between pickup-delivery demands (pd-pairs). Two different categories of splitting strategies are proposed to solve the SDOPDPSTRP: split the demands before the calculation and split the demands during the calculation. Two Multi-Start Variable Neighborhood Descent (a MS_VND originating from the other literature and a new MS_VND' IN developed in this article) and seven neighborhood operators are proposed for these two splitting strategies to solve the SDOPDPSTRP. The results show that Approach III outperforms Approach I and Approach II in terms of average solutions with the same algorithm termination conditions and in terms of time efficiency, which has great practical significance for real-life transport organizations.

This paper proposes a discrete-time, distributed algorithm for multi-agent networks to achieve the minimum l1-norm solution to a group of linear equations known to possess a family of solutions. We assume each agent in the network knows only one equation and can communicate with only its neighbors. The algorithm is developed based on a combination of the projection-consensus idea and the sub-gradient descent method. Given the underlying network graph to be directed and strongly connected, we prove that the algorithm enables all agents to achieve a common minimum l1-norm solution. The major difficulty to be dealt with is the non-smooth nature of the norm and the lack of strict convexity of the associated relevant performance index.

This paper proposes a new class of accelerated conjugate-gradient-like algorithms for solving large scale unconstrained optimization problems, which combine the idea of accelerated adaptive Perry conjugate gradient algorithms proposed by Andrei (2017) with the modified secant condition and the nonmonotone line search technique. An attractive property of the proposed methods is that the search direction always provides sufficient descent step which is independent of the line search used and the convexity of objective function. Under common assumptions, it is proven that the proposed methods possess global convergence for nonconvex smooth functions, and R-linear convergence for uniformly convex functions. The numerical experiments show the efficiency of the proposed method in practical computations.

In this paper, we suggested a modified conjugate gradient method for training neural network which assurance the descent and the sufficient descent conditions. The global convergence of our proposed method has been studied. Finally, the test results present that, in general, the modified method is more superior and efficient when compared to other standard conjugate gradient methods

In this paper, the distributed optimization problem over multi-cluster networks is considered. Different from the existing works, this paper studies the optimization algorithm under uncoordinated step sizes. More specifically, by combining a random sleep strategy and the round-robin communication among clusters, a new hierarchical algorithm is developed to solve the considered problem. In the proposed algorithm, the random sleep strategy enables each agent to independently choose either performing the projected subgradient descent, or keeping the previous estimate by a Bernoulli decision, based on which the step size of each agent is selected as an uncoordinated form that only relates to the independent Bernoulli decision variable. Technically, by introducing a key definition on the algorithm history, it is proven that the estimates of the proposed algorithm can converge to the optimal solution even with the uncoordinated step sizes. In addition, we also study the convergence performance of the proposed algorithm with simpler constant step sizes. In this case, it is proven that the random sleep strategy can efficiently improve the convergence accuracy of the algorithm. Finally, the theoretical findings are verified via simulation examples.

The Column Subset Selection Problem (CSSP) and the Nystrom method are among the leading tools for constructing interpretable low-rank approximations of large datasets by selecting a small but representative set of features or instances. A fundamental question in this area is: what is the cost of this interpretability, i.e., how well can a data subset of size k compete with the best rank k approximation? We develop techniques which exploit spectral properties of the data matrix to obtain improved approximation guarantees which go beyond the standard worst-case analysis. Our approach leads to significantly better bounds for datasets with known rates of singular value decay, e.g., polynomial or exponential decay. Our analysis also reveals an intriguing phenomenon: the cost of interpretability as a function of k may exhibit multiple peaks and valleys, which we call a multiple-descent curve. A lower bound we establish shows that this behavior is not an artifact of our analysis, but rather it is an inherent property of the CSSP and Nystrom tasks. Finally, using the example of a radial basis function (RBF) kernel, we show that both our improved bounds and the multiple-descent curve can be observed on real datasets simply by varying the RBF parameter.

Con el objetivo de aportar una nueva visión en la resolución de algunos problemas de optimización, este artículo versará acerca de la deducción de un nuevo método numérico en optimización. Para ello, se planteará una modelización genérica de los problemas que se plantean resolver.De este modo, se introducirán los preliminares necesarios para poder proceder con la deducción queposteriormente se hará. En este sentido, el método planteado se enmarcará dentro de los métodosde direcciones factibles, por lo que se introducirán conceptos topológicos sencillos (relacionados conel conjunto admisible) y definiciones relacionadas con las direcciones factibles y de descenso que sepodrán considerar. Posteriormente se procederá con la deducción del algoritmo que presenta esteartículo, procurando que el lector pueda observar el procedimiento de construcción del mismo. Ade-más, se introducirán los aspectos computacionales del método y se propondrán una serie de códigosque permitirán implementar el algoritmo en un lenguaje de programación, en el caso de este artículose utilizará MatLab. Finalmente se mostrarán los resultados obtenidos al ejecutar el método junto conotros métodos ya conocidos para poder comparar los resultados obtenidos. De esta forma, se finali-zará aportando las conclusiones más relevantes y las futuras líneas de investigación para mejorar elmétodo presentado.

The double descent curve is one of the most intriguing properties of deep neural networks. It contrasts the classical bias-variance curve with the behavior of modern neural networks, occurring where the number of samples nears the number of parameters. In this work, we explore the connection between the double descent phenomena and the number of samples in the deep neural network setting. In particular, we propose a construction which augments the existing dataset by artificially increasing the number of samples. This construction empirically mitigates the double descent curve in this setting. We reproduce existing work on deep double descent, and observe a smooth descent into the overparameterized region for our construction. This occurs both with respect to the model size, and with respect to the number epochs.

Three-term conjugate gradient methods have attracted much attention for large-scale unconstrained problems in recent years, since they have attractive practical factors such as simple computation, low memory requirement, better descent property and strong global convergence property. In this paper, a hybrid three-term conjugate gradient algorithm is proposed and it owns a sufficient descent property, independent of any line search technique. Under some mild conditions, the proposed method is globally convergent for uniformly convex objective functions. Meanwhile, by using the modified secant equation, the proposed method is also global convergence without convexity assumption on the objective function. Numerical results also indicate that the proposed algorithm is more efficient and reliable than the other methods for the testing problems.

In this paper, we present a new modified inertial forward–backward algorithm for finding a common solution of the quasi-variational inclusion problem and the variational inequality problem in a q-uniformly smooth Banach space. The proposed algorithm is based on descent, splitting and inertial ideas. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to the unique solution of the abovementioned problems. Numerical examples are also given to demonstrate our results.

Refined product distribution is an application of the Multi-compartment Vehicle Routing Problem (MCVRP), which simultaneously considers the vehicle routing, the assignment of heterogonous vehicles and loading policies of multi-compartment. First, we develop an optimization model with the objective of delivering on-time and minimizing transportation cost. Then we propose a Multi-objective Variable Neighborhood Descent Algorithm (MOVND), where the [Formula: see text]-constraint method transforms the original problem into a series of sub-problems of single objective with constraints. Finally, the efficiency of the proposed algorithm is verified by conducting a large number of small and large instances. Mainly including (i) compared with the classical NSGA-2 algorithm for multi-objective VRP, MOVND provides better performance in terms of convergence, spread and distribution; (ii) the multi-compartment vehicles are able to carry a variety of products simultaneously, which can improve the effective utilization of vehicle space and meet the needs of different customers in a single transportation. In addition, heterogonous vehicles that take full advantage of the characteristics of different vehicles are superior to homogeneous ones in terms of operating cost in the practice of the refined product distribution.

We give an introduction to the MARS toolchain for formal modeling and verification of hybrid systems. It consists of translators from Simulink/Stateflow models to Hybrid Communicating Sequential Processes (HCSP), and tools for simulation, code generation, and deductive verification of an HCSP model. We apply the toolchain to model the descent guidance control phase of the recently launched Tianwen I mars lander, and verify that it correctly controls the velocity of the lander.

Active inference is a normative framework for explaining behaviour under the free energy principle -- a theory of self-organisation originating in neuroscience. It specifies neuronal dynamics for state-estimation in terms of a descent on (variational) free energy -- a measure of the fit between an internal (generative) model and sensory observations. The free energy gradient is a prediction error -- plausibly encoded in the average membrane potentials of neuronal populations. Conversely, the expected probability of a state can be expressed in terms of neuronal firing rates. We show that this is consistent with current models of neuronal dynamics and establish face validity by synthesising plausible electrophysiological responses. We then show that these neuronal dynamics approximate natural gradient descent, a well-known optimisation algorithm from information geometry that follows the steepest descent of the objective in information space. We compare the information length of belief updating in both schemes, a measure of the distance traveled in information space that has a direct interpretation in terms of metabolic cost. We show that neural dynamics under active inference are metabolically efficient and suggest that neural representations in biological agents may evolve by approximating steepest descent in information space towards the point of optimal inference., Comment: 12 pages, 3 figures

With the grown of global concern with environmental issues and incentives on the part of governments, the use of electric vehicles (EVs) by companies has increased. By joining the interest of governments on minimizing the costs of building the recharging infrastructure and of companies on minimizing their transport costs with the adoption of electric vehicles in their fleet, a bilevel optimization problem arises. An approach that combines the Variable Neighborhood Search (VND) with an Ant Colony Optimization (ACO) to solve this bilevel model is proposed here. The results of the computational experiments using different benchmark instances indicate the superior performance of the proposed approach, mainly in the allocation of the charging infrastructure (objective of the leader). Also, the proposal found better solutions than those from the literature while maintains the percentage of electric vehicles that compose the fleet above the required limit.

In some previous works, two of the authors introduced a strategy to develop high-order well-balanced numerical methods for 1d systems of balance laws. There, a strategy which allows us to modify any standard reconstruction operator in order to be well-balanced was also described. This strategy involves a nonlinear problem at every cell, at every time step, that consists in finding the stationary solution whose average is the given cell value. Our goal is to present a general efficient implementation that can be applied to any system of balance laws by interpreting these nonlinear problems as control problems that are rewritten in functional form. Newton’s and descent methods are applied and compared. Applications to the Burgers’ equation with a nonlinear source term and to the 1d shallow water model are finally shown.

Sparse channel estimation for massive multiple-input multiple-output systems has drawn much attention in recent years. The required pilots are substantially reduced when the sparse channel state vectors can be reconstructed from a few numbers of measurements. A popular approach for sparse reconstruction is to solve the least-squares problem with a convex regularization. However, the convex regularizer is either too loose to force sparsity or lead to biased estimation. In this paper, the sparse channel reconstruction is solved by minimizing the least-squares objective with a nonconvex regularizer, which can exactly express the sparsity constraint and avoid introducing serious bias in the solution. A novel algorithm is proposed for solving the resulting nonconvex optimization via the difference of convex functions programming and the gradient projection descent. Simulation results show that the proposed algorithm is fast and accurate, and it outperforms the existing sparse recovery algorithms in terms of reconstruction errors., Comment: 2020 IEEE Global Communications Conference

This paper deals with the Green Electric Vehicle Routing Problem with Time Window and Mixed Fleet and presents a Mixed Integer Linear Programming formulation for it. Initially, we applied the CPLEX solver in this formulation. Then, to reduce the computational time, we used Local Branching and Variable Neighborhood Descent Branching (VNDB) methods. We did computational experiments with a simple adaptation of the 100-customers Solomon’s benchmark instances. The results showed that the three solution strategies reached the optimal solution. However, the running time of the VNDB is considerably smaller than those required by the other two solution methods. Therefore, this fact proves that the VNDB is the more efficient technique in the tested scenario.

The conjugate gradient is a useful tool in solving large- and small-scale unconstrained optimization problems. In addition, the conjugate gradient method can be applied in many fields, such as engineering, medical research, and computer science. In this paper, a convex combination of two different search directions is proposed. The new combination satisfies the sufficient descent condition and the convergence analysis. Moreover, a new conjugate gradient formula is proposed. The new formula satisfies the convergence properties with the descent property related to Hestenes–Stiefel conjugate gradient formula. The numerical results show that the new search direction outperforms both two search directions, making it convex between them. The numerical result includes the number of iterations, function evaluations, and central processing unit time. Finally, we present some examples about image restoration as an application of the proposed conjugate gradient method.

Logistic regression is a classic classification method in machine learning. Classical logistic regression uses a general gradient descent method to solve the best parameters of the loss function, but it is easy to fall into the dilemma of local extremes. The fractional step descent method cannot converge to the exact extreme point, and there is always a deviation, so we use a variable fractional step descent method to simulate a new type of iteration to ensure the global convergence of the optimization algorithm. In this paper, the new variable fractional gradient descent method is used to solve the optimal parameters of the loss function, which overcomes the limitation of the gradient function on the step size of the loss function. In this paper, logistic regression and variable fractional step descent methods are used to deal with the problem of data dimensionality reduction to verify the effectiveness of the algorithm after optimization.

Gradient descent is a widely used optimization method. The adjustment of the learning rate is an important factor in improving its performance, and many researchers have investigated methods for automatically adjusting the learning rate. One such method, hypergradient descent, automatically adjusts the learning rate by using gradient descent. However, it introduces the “learning rate of the learning rate,” and an appropriate value for the learning rate of the learning rate must be chosen in order to effectively adjust the learning rate. We investigated the use of two datasets and two optimization methods for doing this and achieved an effective adjustment of the learning rate when the objective function was convex and $L$ -smooth.

Descent & landing (D&L) on small planetary bodies are scientifically rewarding exploration missions but they are technically challenging due to the complex and poorly-known environment around those bodies. The standard guidance synthesis approach considers nominal conditions and applies optimal control theory to obtain guidance law gains, followed by intensive verification and validation. In this article, it is shown that the standard approach may yield gains that are not optimal once dispersions (and/or other optimality metrics) are taken into account and a tuning approach is then proposed based on a priori methodological system assessment. The proposed approach employs systematic high-fidelity simulations to generate trade-off maps. These maps can be generated by on ground operators based on the best estimated conditions and uploaded to the spacecraft as it approaches the target. The proposed systematic guidance tuning and resulting maps also provide a valuable understanding of the system dynamics towards the application of other industry-oriented tools such as structured Η∞ optimisation. It is shown that the proposed tuning enables propellant consumption reductions of around 40% compared to state-of-practice gain selections.

This paper presents an optimal control problem to analyze the efficacy of counter-terrorism tactics. We present an algorithm that efficiently combines the Minimum Principle of Pontryagin, the shooting method and the cyclic descent of coordinates. We also present a result that allows us to know a priori the steady state solutions. Using this technique we are able to choose parameters that reach a specific solution, of which there are two. Numerical examples are presented to illustrate the possibilities of the method. Finally, we study the sufficient conditions for optimality and suggest an improvement on the functional which also guarantees local optimality.

In the paper we propose an accelerated directional search method with non-euclidian prox-structure. We consider convex unconstraint optimization problem in $\mathbb{R}^n$. For simplicity we start from the zero point. We expect in advance that 1-norm of the solution is close enough to its 2-norm. In this case the standard accelerated Nesterov's directional search method can be improved. In the paper we show how to make Nesterov's method $n$-times faster (up to a $\log n$-factor) in this case. The basic idea is to use linear coupling, proposed by Allen-Zhu & Orecchia in 2014, and to make Grad-step in 2-norm, but Mirr-step in 1-norm. We show that for constrained optimization problems this approach stable upon an obstacle., in Russian, 13 pages, 3 figures

Memoryless quasi-Newton methods are studied for solving large-scale unconstrained optimization problems. Recently, memoryless quasi-Newton methods based on several kinds of updating formulas were proposed. Since the methods closely related to the conjugate gradient method, the methods are promising. In this paper, we propose a memoryless quasi-Newton method based on the Broyden family with the spectral-scaling secant condition. We focus on the convex and preconvex classes of the Broyden family, and we show that the proposed method satisfies the sufficient descent condition and converges globally. Finally, some numerical experiments are given.

This paper considers a refinement problem for the flight altitude, speed, and flight-path angle of an aircraft during aerodynamic deceleration in the atmosphere of Mars using inaccurate discrete g-load measurements. For solving this problem, several suboptimal discrete nonlinear filters of higher accuracy than the extended Kalman filter based on the Taylor linearization procedure with static (Gaussian) linearization of the nonlinearities are applied and compared with each other. The equations of these filters are derived, the Gaussian moments of their structural functions are calculated, and the resulting accuracy of these filters is analyzed in statistical terms.

This paper present algorithms for solving a single machine scheduling problem to minimize the sum of total completion times, total tardiness,maxim-um tardiness,and maximum earliness.The single machine total tardiness problem is already NP-hard, so the consider problem is strongly NP-hard, and several algorithms are used to solve it. Branch and bound algorithmwith dominance ruleand local search algorit- hms are proposed for the problem. For the Branch and bound algorithm results- show that using dominance rule improve the performance of the algorithm in both computation times and optimal values,but it need longer times.Thus we tackle the problemof large sizes with local search algorit- hms descent method, simulated annealing and tabusearch. The perfomance of these algorithms is evaluated on a large set of test problems and the results are compared.The computational results show that simulated annealing algorithm and Tabu search algorithm are better than Descent method with preference to simulated annealing algorithm,and show that the three algorithms find optimal or near optimal solutions inreasonable times.

Focusing on the trajectory curvature, this paper presents an innovative and analytical guidance law for the construction of geometrically convex trajectories. Moving along such trajectories, the lander can increase the probability of a safe landing in hazardous terrains. Initially, the curvature theorems of the powered descent trajectories are developed. In these theorems, the inner relationship between trajectory curvature and lander states is revealed, and the state constraints for a geometrically convex trajectory are derived. Next, the trajectory curvature guidance is developed in an analytical formulation by satisfying the state constraints for a convex trajectory, and the selection of the key guidance parameters is investigated. Finally, the performance of the trajectory curvature guidance is analyzed in detail, illustrating its superior hazard avoidance and the camera’s field of view.

A direct multiple shooting (MS) method is implemented to solve optimal control problems (OCP) in the Mayer form. The use of an MS method gives rise to the so-called ‘continuity conditions’ that must be satisfied together with general algebraic equality and inequality constraints. The resulting finite nonlinear optimization problem is solved by a first-order descent method based on the filter methodology. In the equivalent tri-objective problem, the descent method aims to minimize the objective function, the violation of the ‘continuity conditions’ and the violation of the algebraic constraints simultaneously. The numerical experiments carried out with different types of benchmark OCP are encouraging., We acknowledge the financial support of CIDEM, R&D unit funded by the FCT - Portuguese Foundation for the Development of Science and Technology, Ministry of Science, Technology and Higher Education, under the Project UID/EMS/0615/2019, and FCT within the Projects Scope: UID/CEC/00319/2019 and UID/MAT/00013/2013.

In this paper, we propose a hybrid conjugate gradient (CG) method based on the approach of convex combination of Fletcher–Reeves (FR) and Polak–Ribiere–Polyak (PRP) parameters, and Quasi-Newton’s update. This is made possible by using self-scaling memory-less Broyden’s update together with a hybrid direction consisting of two CG parameters. However, an important property of the new algorithm is that, it generates a descent search direction via non-monotone type line search. The global convergence of the algorithm is established under appropriate conditions. Finally, numerical experiments on some benchmark test problems, demonstrate the effectiveness of the proposed algorithm over some existing alternatives.

Many researchers are interested for developed and improved the conjugate gradient method for solving large scale unconstrained optimization problems. In this work a new parameterøn will be presented as a convex combination between RMIL and MMWU. The suggestion method always produces a descent search direction at each iteration. Under Strong Wolfe Powell (SWP) line search conditions, the global convergence of the proposed method is established. The preliminary numerical comparisons with some others CG methods have shown that this new method is efficient and robust in solving all given problems.