Search

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.

Nonlinear conjugate gradient methods are often used to solve large-scale unconstrained optimization problems owing to their lower memory requirement and less computation cost. In this paper, we propose a new Polak-Ribiere-Polyak type conjugate gradient method, which satisfies the sufficient descent condition independent of any line search. A remarkable property about this method is its strong global convergence with the standard Wolfe line search conditions as well as the standard Armijo line search strategy without convexity assumption of the objective function. The numerical results demonstrating the efficiency of the proposed method are reported.

In this paper, we examine the behavior of ideal-adic separatedness and completeness under certain ring extensions using trace map. Then we prove that adic completeness of a base ring is hereditary to its ring extension under reasonable conditions. We aim to give many results on ascent and descent of certain ring theoretic properties under completion. As an application, we give conceptual details to the proof of the almost purity theorem for Witt-perfect rings by Davis and Kedlaya. Witt-perfect rings have the advantage that one does not need to assume that the rings are complete and separated.

By computing the completely bounded norm of the flip map on the Haagerup tensor product $C_0 Y_1\otimes_{C_0 X} C_0 Y_2$ associated to a pair of continuous mappings of locally compact Hausdorff spaces $Y_1\rightarrow X\leftarrow Y_2$, we establish a simple characterisation of the Beck-Chevalley condition for base change of operator modules over commutative $C^*$-algebras, and a descent theorem for continuous fields of Hilbert spaces., Comment: 10 pages. Minor changes to the exposition; corrected an error in Remarks 6 not affecting the rest of the paper

It is known that the conjugate gradient method is still a popular method for many researchers who are focused in solving the large-scale unconstrained optimization problems and nonlinear equations because the method avoids the computation and storage of some matrices so the memory's requirements of the method are very small. In this work, a modified Perry conjugate gradient method which fulfills a global convergence with standard assumptions is shown and analyzed. The idea of new method is based on Perry method by using the equation which is founded via Powell in 1978. The weak Wolfe–Powell search conditions are used to choose the optimal line search, under the line search and suitable conditions, we prove both descent and sufficient descent conditions. In particular, numerical results show that the new conjugate gradient method is more effective and competitive when compared to other of standard conjugate gradient methods including: - CG- Hestenes and Stiefel (H/S) method, CG-Perry method, CG- Dai and Yuan (D/Y) method. The comparison is completed under a group of standard test problems with various dimensions from the CUTEst test library and the comparative performances of the methods are evaluated by total the number of iterations and the total number of function evaluations.

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.

This paper is a continuation of our previous one under the same title. In both articles, we study the hyperelliptic curves [Formula: see text] defined over [Formula: see text], and their Jacobians [Formula: see text] (without loss of generality a is a nonzero 8th power free integer). Previously, we considered the case when the polynomial [Formula: see text] is irreducible in [Formula: see text] and obtained (under certain conditions on the quartic field [Formula: see text]) upper bounds for the [Formula: see text]; in particular, we found infinite subfamilies of [Formula: see text] with rank zero. Now we consider all cases when [Formula: see text] is reducible in [Formula: see text] and prove analogous results. First we obtain (under mild conditions on some quadratic fields) upper bounds for the ranks in a rather general situation, then we restrict to a several infinite subfamilies of [Formula: see text] (when 2a has two primes divisors) and get the best possible bounds or even the exact value of rank (if it is zero). We deduce as conclusions the complete lists of rational points on [Formula: see text] in such cases.

Two improved nonlinear conjugate gradient methods are proposed by using the second inequality of the strong Wolfe line search. Under usual assumptions, we proved that the improved methods possess the sufficient descent property and global convergence. By testing the unconstrained optimization problems which taken from the CUTE library and other usual test collections, some large-scale numerical experiments for the presented methods and their comparisons are executed. The detailed results are listed in tables and their corresponding performance profiles are reported in figures, which show that our improved methods are superior to their comparisons.

We generalise sheaf models of intuitionistic logic to univalent type theory over a small category with a Grothendieck topology. We use in a crucial way that we have constructive models of univalence, that can then be relativized to any presheaf models, and these sheaf models are obtained by localisation for a left exact modality. We provide first an abstract notion of descent data which can be thought of as a higher version of the notion of prenucleus on frames, from which can be generated a nucleus (left exact modality) by transfinite iteration. We then provide several examples., Simplified the definition of lex operation, simplified the encoding of the homotopy limit and remark that the homotopy descent data is a lex modality without using higher inductive types

In two pre-registered studies, we leveraged recent advances to disposition theory to examine whether character judgments are relative. We used a Pilot Study to develop a moral continuum of behaviors for a hypothetical television series. We referenced our established moral continuum to create behavioral sequences that represented two characters descending into immorality. We manipulated whether one or both characters were present in the narrative. The simultaneous presence of both characters polarized participants’ moral evaluations of character behavior, categorization of the characters as heroic/villainous, and character liking. Our findings substantiate the systematic effects that character interdependence has on disposition formation. An improved understanding of narrative context can specify when between- and within-character comparisons occur and what effects character interdependence has on disposition theory’s processes. We discuss how narrative schemas, character schemas, and character networks can serve as the elements for explicating the role of narrative context in disposition theory.

Given an integer n ≥1, we provide a complete description of all bijective bicontinuous maps, on the algebra of all bounded linear operators acting on an in nite-dimensional complex or real Banach space, that preserve the difference of Drazin invertible operators of index non-greater than n in both directions. Key words: Nonlinear preserver problems, Drazin inverse, ascent, descent.

Recently, Jian, Han and Jiang proposed a descent hybrid conjugate gradient method which is globally convergent without convexity assumption on the objective function, being also sensibly promising in computational point of view. Here, we develop one-parameter descent extensions of the method based on the Dai–Liao approach. We show that one of the given methods satisfies the sufficient descent condition when the parameter is chosen properly. Also, we establish global convergence of the method without convexity assumption. At last, practical merits of the methods are investigated by numerical experiments on a set of CUTEr test functions as well as the signal processing problems. The results show computational efficiency of the proposed methods.

Let $\mathcal{P}_s(n)$ denote the $n$th $s$-gonal number. We consider the equation \[\mathcal{P}_s(n) = y^m, \] for integers $n,s,y,$ and $m$. All solutions to this equation are known for $m>2$ and $s \in \{3,5,6,8,20 \}$. We consider the case $s=10$, that of decagonal numbers. Using a descent argument and the modular method, we prove that the only decagonal number $>1$ expressible as a perfect $m$th power with $m>1$ is $\mathcal{P}_{10}(3) = 3^3$., 5 pages. Some typos corrected

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.

The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations, called alternating Eulerian polynomials, are unimodal via a five-term recurrence relation. We also find a quadratic recursion for the alternating major index q-analog of the alternating Eulerian polynomials. As an interesting application of this quadratic recursion, we show that $$(1+q)^{\lfloor n/2\rfloor }$$ divides $$\sum _{\pi \in {{\mathfrak {S}}}_n}q^{\mathrm{altmaj}(\pi )}$$ , where $${{\mathfrak {S}}}_n$$ is the set of all permutations of $$\{1,2,\ldots ,n\}$$ and $$\mathrm{altmaj}(\pi )$$ is the alternating major index of $$\pi $$ . This leads us to discover a q-analog of $$n!=2^{\ell }m$$ , m odd, using the statistic of alternating major index. Moreover, we study the $$\gamma $$ -vectors of the alternating Eulerian polynomials by using these two recursions and the cd-index. Further intriguing conjectures are formulated, which indicate that the alternating descent statistic deserves more work.

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.

Let φ and ψ be regular quadratic forms over a field F of characteristic different from 2. We say that ψ is a quasisubform of φ if there is a∈F* such that aψ is a subform of φ. Let L/F be an odd deg...

We use descent theoretic methods to solve the homotopy limit problem for Hermitian $K$-theory over quasi-compact and quasi-separated base schemes. As another application of these descent theoretic methods, we compute the cellular Picard group of 2-complete Hermitian $K$-theory over $\mathop{Spec}(\mathbb{C})$, showing that the only invertible cellular spectra are suspensions of the tensor unit., Comment: v4: results made unconditional, assumptions on base scheme weakened. Version to appear in Annals of K-theory

This study aims at describing the concept of al-amwal in the Qur'an which is referred to as treasure. This research focuses more on the concept of property and property ownership in the Qur'an, property as slander (trials), worldly treasures (al-hayah al-dunya), property as a means of doing good. Descriptive research method with qualitative approach is employed in this study. This present study, is a library study using maudhu’i method. Maudhu’i method is a method seeks to find the answer in the Qur'an by collecting ayahs of the Qur'an that have similar purpose, which together discuss a particular topic / title and publish those ayah in accordance with the time of thier descent in accordance with the causes of their descent, then pay attention to the ayahs with explanation, description and relationship with other ayahs. The results show that wealth (wealth) is a primary need for man to support his life in this world. The Qur'an reminds that property must be obtained through lawful manners as stipulated by the shariah law. In fact the Qur'an and the Sunnah clearly explain that property should not be transferred from one person to another through unjustified ways. In stead, it should be transferred based on the sharia, such as through trades or other lawful transactions. The use of property in Islamic teachings must always be in devotion to God and used in the framework of taqqarub (to get closer) to Allah.

The purpose of this paper is to present in linear spaces some results for new notions called A-left (resp., A-right) ascent and A-left (resp., A-right) descent of linear operators (where A is a given operator) which generalize two important notions in operator theory: ascent and descent. Moreover, if A is a positive operator, we obtain several properties of ascent and descent of an operator in semi-Hilbertian spaces. Some basic properties and many results related to the ascent and descent for a linear operator on a linear space Kaashoek (Math Ann 172:105–115, 1967), Taylor (Math Ann 163:18–49, 1966) are extended to these notions. Some stability results under perturbations by compact operators and operators having some finite rank power are also given for these notions.

This paper considers sufficient descent Riemannian conjugate gradient methods with line search algorithms. We propose two kinds of sufficient descent nonlinear conjugate gradient method and prove that these methods satisfy the sufficient descent condition on Riemannian manifolds. One is a hybrid method combining a Fletcher–Reeves-type method with a Polak–Ribiere–Polyak-type method, and the other is a Hager–Zhang-type method, both of which are generalizations of those used in Euclidean space. Moreover, we prove that the hybrid method has a global convergence property under the strong Wolfe conditions and the Hager–Zhang-type method has the sufficient descent property regardless of whether a line search is used or not. Further, we review two kinds of line search algorithm on Riemannian manifolds and numerically compare our generalized methods by solving several Riemannian optimization problems. The results show that the performance of the proposed hybrid methods greatly depends on the type of line search used. Meanwhile, the Hager–Zhang-type method has the fast convergence property regardless of the type of line search used.

More attention has been devoted to providing evolutionary accounts of the development of beliefs, or belief-like states, than for desires or preferences. Here I articulate and defend an evolutionar...

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.

Linear programs (LPs) can be solved by polynomially many moves along the circuit direction improving the objective the most, so-called deepest-descent steps (dd-steps). Computing these steps is NP-hard (De Loera et al. (2019)), a consequence of the hardness of deciding the existence of an optimal circuit-neighbor (OCNP) on LPs with non-unique optima. We prove OCNP is easy under the promise of unique optima, but already O ( n 1 − e ) -approximating dd-steps remains hard even for totally unimodular n -dimensional 0/1-LPs with a unique optimum. We provide a matching n -approximation.

In this paper, a family of coordinate majorization descent algorithms are proposed for solving the nonconvex penalized learning problems including SCAD and MCP estimation. In the coordinate majoriz...

The semiperimeter s, inradius r and circumradius R are the symmetric invariants of a triangle. In this short note, we complement previous work of Hirakawa and Matsumura by determining all pairs (up to similitude) consisting of a rational right angled triangle and a rational isosceles triangle having two corresponding symmetric invariants equal. In the proof of our main theorems, we determine the set of rational points on two hyperelliptic curves of genus 3. The first of these can be settled via a classical descent argument, whereas the second one is proved using ‘‘elliptic curve Chabauty’’.

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.

We apply an explicit construction of a simplicial powering in dg-categories, due to Holstein (Properness and simplicial resolutions for the model category dgCat, 2014. arXiv:1403.4381 ) and Arkhipov and Poliakova (Homol Homotopy Appl 22(2):151–162, 2019), as well as our own results on homotopy ends (Arkhipov and Orsted in Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories, 2018. arXiv:1807.03266 ), to obtain an explicit model for the homotopy limit of a cosimplicial system of dg-categories. We apply this to obtain a model for homotopy descent in terms of $$\mathrm {A}_{\infty }$$ -comodules, proving a conjecture by Block et al. (Homol Homotopy Appl 19(2):343–371, 2017) in the process.

Let $K$ be a $C_1$-field of any characteristic and $X$ a projective variety over $K$. In this article we prove that for a finite Galois extension $L$ of $K$, a simple sheaf with covering datum on $X \times_K L$ descends to a simple sheaf on $X$. As a consequence, we show that there is a $1-1$ correspondence between the set of geometrically stable sheaves on $X$ with fixed Hibert polynomial $P$ and the set of $K$-rational points of the corresponding moduli space., Appeared in the Proceedings of the ICM 2018 satellite 'Moduli spaces in Algebraic Geometry and Applications'

In this paper, by linearly combining the numerator and denominator terms of the Dai-Liao (DL) and Bamigbola-Ali-Nwaeze (BAN) conjugate gradient methods (CGMs), a general form of DL-BAN method has been proposed. From this general form, a new hybrid CGM, which was found to possess a sufficient descent property is generated. Numerical experiment was carried out on the new CGM in comparison with four existing CGMs, using some set of large scale unconstrained optimization problems. The result showed a superior performance of new method over majority of the existing methods.

The present study examined the Interpersonal Theory of Suicide (IPT) within a culturally relevant framework by simultaneously examining familismo values to explain suicidal ideation among Mexican descent emerging adults. With a sample of 249 Mexican descent emerging adults (age range: 18–25 years; M = 19.17; SD = 1.53; 65.5% women), the present study found that there was a significant interaction between perceived burdensomeness (PB) and familismo-subjugation of self values. These findings provide evidence that in the context of perceiving one’s self as being a burden on the family, having higher familismo subjugation of self-values puts one at higher risk for suicidal ideation than if they have lower subjugation values. These findings highlight that when dissonance occurs between one’s values and their lived experience, a higher risk of suicidal ideation occurs for Mexican descent emerging adults.

The Spectral conjugate gradient (SCG) methods are among the efficient variants of CG algorithms which are obtained by combining the spectral gradient parameter and CG parameter. The success of SCG methods relies on effective choices of the step-size $\alpha _{k}$ and the search direction $d_{k}$ . This paper presents an SCG method for unconstrained optimization models. The search directions generated by the new method possess sufficient descent property without the restart condition and independent of the line search procedure used. The global convergence of the new method is proved under the weak Wolfe line search. Preliminary numerical results are presented which show that the method is efficient and promising, particularly for large-scale problems. Also, the method was applied to solve the robotic motion control problem and portfolio selection problem.

For the hyperelliptic curve C_p with equation y^2=x(x-2p)(x-p)(x+p)(x+2p) with p a prime number, we discuss bounds for the rank of its Jacobian over Q, find many cases having 2-torsion in the associated Shafarevich-Tate group, and we present some results on rational points of C_p., 17 pages

The article considers the problem of optimal control of an object described by a linear nonstationary system and with a piecewise affine quality functional. The problem is examined in Mayer’s form with both free and partially fixed right endpoints. Piecewise continuous and bounded controls that lie in some parallelepiped at each moment of time are admissible. The standard discretization of the original system and the control parametrization are used, some convergence theorems of the discrete problem solution to the continuous problem solution are presented. Further, for the obtained discrete system, the necessary and sufficient minimum conditions are written out in terms of the codifferential, the method of the modified codifferential descent is applied to it, which guarantees to find the global minimum of this problem in a finite number of steps. The proposed algorithm is illustrated with examples.

This paper engages with a specific image: Darwin's tree of the primates. Although this diagram was sketched in ink on paper in 1868, it did not make it into the publication ofThe Descent of Man(1871). This may seem all the more in need of an explanation because, as Adrian Desmond and James Moore have shown, Darwin strongly relied on the notion of familial genealogy in the development of his theory of organismic evolution, or rather descent. However, Darwin expressed scepticism towards visualizations of phylogenies in correspondence with Ernst Haeckel and in fact also inDescent, considering such representations at once too speculative and too concrete. An abstraction such as a tree diagram left little room to ponder possibilities or demarcate hypotheses from evidence. I thus bring Darwin's primate tree into communication with his view on primate and human phylogeny as formulated inDescent, including his rejection of polygenism. I argue that considering the tree's inherent teleology, as well as its power to suggest species status of human populations and to reify ‘racial’ hierarchies, the absence of the diagram inThe Descent of Manmay be a significant statement.

Let ${\mathcal L}/{\mathcal K}$ be a finite Galois extension and let $X$ be an affine algebraic variety defined over ${\mathcal L}$. Weil's Galois descent theorem provides necessary and sufficient conditions for $X$ to be definable over ${\mathcal K}$, that is, for the existence of an algebraic variety $Y$ defined over ${\mathcal K}$ together with a birational isomorphism $R:X \to Y$ defined over ${\mathcal L}$. Weil's proof does not provide a method to construct the birational isomorphism $R.$ The aim of this paper is to give an explicit construction of $R$., Comment: 11 pages

Most studies of the American reception of Darwin have focused on theOrigin. TheDescent of Man, however, was even more widely read and discussed, especially by those outside the emerging scientific establishment. This essay maps the varied, popular and radical responses to theDescentand suggests that these unauthorized readers helped shape the formation of American scientific institutions (by encouraging scientists to close ranks), as well as ordinary Americans’ perceptions of gender and sex. I argue that the radical – freethinkers, socialists and feminists – embrace of sexual selection theory provides one explanation for naturalists’ scepticism of the theory.

Shape gradients have been widely used in numerical shape gradient descent algorithms for shape optimization. The two types of shape gradients, i.e., the distributed one and the boundary type, are e...

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.

Given a convex objective function on a Banach space, which is Lipschitz on bounded sets, we consider the class of regular vector fields introduced in our previous work on descent methods. We analyz...

We implement two-cover descent for plane quartics over Q with all 28 bitangents rational and show that on a significant collection of test cases, it resolves the existence of rational points. We also review a classical description of the relevant moduli space and use it to generate examples. We observe that local obstructions are quite rare for such curves, and only seem to occur in practice at primes of good reduction. In particular, having good reduction at 11 implies having no rational points. We also gather numerical data on two-Selmer ranks of Jacobians of these curves, which suggests that these often have non-trivial Tate-Shafarevich groups. We implement two-cover descent for plane quartics over Q with all 28 bitangents rational and show that on a significant collection of test cases, it resolves the existence of rational points. We also review a classical description of the relevant moduli space and use it to generate examples. We observe that local obstructions are quite rare for such curves and only seem to occur in practice at primes of good reduction. In particular, having good reduction at 11 implies having no rational points. We also gather numerical data on two-Selmer ranks of Jacobians of these curves, providing evidence these behave differently from those of general abelian varieties due to the frequent presence of an everywhere locally trivial torsor., 15 pages; Some minor improvements to algorithm and rank data analysis

The article clarifies a well-known man of literature who showed backbone by his tolerance during the hard times of repression in Mongolia. He was the last nobleman descent from the Golden family of Chinggis Khaan. However, he is famous for his literary writings, especially the narrative story 'Soyol Erdene' as his own biography.

In this paper, we propose a new conjugate gradient method for solving nonlinear unconstrained optimization. The new method consists of three parts, the first part of them is the parameter of Hestenes-Stiefel (HS). The proposed method is satisfying the descent condition, sufficient descent condition and conjugacy condition. We give some numerical results to show the efficiency of the suggested method.

We consider a non-stationary mixed variational inequality problem involving an integrable mapping and a convex function, where only approximation sequences are known instead of exact values of the cost mapping and function, and feasible set. We apply a descent method and partial penalization to prove the convergence is attained without concordance of penalty, accuracy, and approximation parameters under coercivity type conditions.

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.