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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...

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...

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...

To solve a large-scale unconstrained optimization problem, in this paper we propose a class of spectral three-term conjugate gradient methods. We indicate that the proposed class, in fact, generate...

This paper begins with a brief review of affine invariance and its significance for iterative algorithms. It then explores the existence of affine invariant descent directions for unconstrained min...

Given a Lipschitz convex and coercive objective function on a Banach space, we revisit the class of regular vector fields introduced in our previous work on descent methods. Taking into account com...

In this paper, based on the numerical efficiency of Hestenes–Stiefel (HS) method, a new modified HS algorithm is proposed for unconstrained optimization. The new direction independent of the line search satisfies in the sufficient descent condition. Motivated by theoretical and numerical features of three-term conjugate gradient (CG) methods proposed by Narushima et al., similar to Dai and Kou approach, the new direction is computed by minimizing the distance between the CG direction and the direction of the three-term CG methods proposed by Narushima et al. Under some mild conditions, we establish global convergence of the new method for general functions when the standard Wolfe line search is used. Numerical experiments on some test problems from the CUTEst collection are given to show the efficiency of the proposed method.

In this paper, we study properties of a special class of ordered median functions, called block-functions. These are ordered median functions which belong to a generating binary (row)-vector of the form called a block vector. The aim of this paper is to explicitly determine the simplicial complexes and all steepest descent and ascent directions of descent and ascent cones of ordered median block-function.

In this paper, we propose a non-monotone line search method for solving optimization problems on Stiefel manifold. The main novelty of our approach is that our method uses a search direction based on a linear combination of descent directions and a Barzilai–Borwein line search. The feasibility is guaranteed by projecting each iterate on the Stiefel manifold through SVD (singular value decomposition) factorizations. Some theoretical results for analysing the algorithm are presented. Finally, we provide numerical experiments for comparing our algorithm with other state-of-the-art procedures. The code is available online. The experimental results show that the proposed algorithm is competitive with other approaches and for particular problems, the computational performance is better than the state-of-the-art algorithms.

We prove that for any finite dimensional representation V of a finite group G of order n the quotient variety G∖ℙ(V) is projectively normal with respect to descent of 𝒪(1)⊗l where l = lcm{1,2,3,4,…,n}. We also prove that for the tautological representation V of the alternating group An the projective variety An∖ℙ(V) is projectively normal with respect to the descent of the above line bundle.

In this paper, we show how a nonlinear scalarization functional can be used in order to characterize several well-known set order relations and which thus plays a key role in set optimization. By means of this functional, we derive characterizations for minimal elements of set-valued optimization problems using a set approach. Our methods do not rely on any convexity assumptions on the considered sets. Furthermore, we develop a derivative-free descent method for set optimization problems without convexity assumptions to verify the usefulness of our results.

A few years ago, the authors proposed a nature-inspired metaheuristic concept, the spiral optimization algorithm, which was inspired by spiral phenomena in nature. The principal idea of the algorit...

We suggest a modification of the descent splitting methods for decomposable composite optimization problems, which maintains the basic convergence properties, but enables one to reduce the computational expenses per iteration and to provide computations in a distributed manner. It consists of making coordinate-wise steps together with a special threshold control.

This article concerns optimal control and stabilization for some Fisher-like models with control acting in a subdomain ω. We investigate the optimal position of ω for some optimal harvesting problems. First, we refer to a logistic model with diffusion. We remember the necessary optimality conditions, and then obtain an iterative method to improve the position of ω for the optimal harvesting effort (for a simplified model without logistic term). Next, we consider the null stabilization for a controlled Fisher model and obtain a descent method to improve the position of ω in order to get a faster stabilization to zero. Numerical tests illustrating the effect of the last method are given. We also studied the null stabilization for a prey-predator system and have reduced it to the study of the null stabilizability for a related Fisher model.

Bregman iterative algorithms have been extensively used for and total variation regularization problems, allowing to obtain simple, fast and effective algorithms. In this paper, three already-available algorithms for geosounding inversion are modified by including them in a Bregman iterative procedure. The resulting algorithms are easy to implement and do not require any optimization package. Modelling results are presented for synthetic and field data, observing better convergence properties than the original versions, avoiding the need of any continuation descent procedure.

International audience; We consider a simplied model of a coupled parabolic PDE-ODE describing heat transfer within buildings. We describe an identication procedure able to reconstruct the parameters of the model. The response of the model is nonlinear with respect to its parameters and the reconstruction of the parameters is achieved by the introduction of a new vectorial descent stepsize, which improves the convergence of the Levenberg-Marquardt minimization algorithm. The new vectorial descent stepsize can have negative and positive entries of different sizes, which fundamentally differs from standard scalar descent stepsize. The new algorithm is proved to converge and to outperform the standard scalar descent strategy. We also propose algorithms for the initialization of the parameters needed by the reconstruction procedure, when no a priori knowledge is available.

An ordered median functions is a continuous piecewise-linear function. It is well known that in finite dimensional spaces every continuous piecewise-linear function admits a max-min representation in terms of its linear functions. An explicit representation of an ordered median function in max-min form is given by the authors and will appear in a forthcoming issue of this journal. Based on this representation, we give a topological classification of ordered median functions through their simplicial complex of ascent (resp. descent) cones.

We consider the monotone composite variational inequality CVI where the underlying mapping is formed as the sum of two monotone mappings. We combine the forward–backward and descent direction ideas together, and thus present the unified algorithmic framework of forward–backward-based descent methods for solving the CVI. A new iterate of such a method is generated by a prediction–correction fashion, where the predictor is yielded by the forward–backward method and then the predictor is corrected by a descent step. We derive some implementable forward–backward-based descent algorithms for some concrete cases of the CVI, and verify their numerical efficiency via preliminary numerical experiments.

In [Zhou, A short note on the global convergence of the unmodified PRP method, Optim. Lett. doi: 10.1007/s11590-012-0511-7, to appear], Zhou showed that the classical unmodified Polak–Ribiere–Polyak PRP nonlinear conjugate gradient method converges globally in the sense that lim infk→∞‖∇fxk‖=0 for nonconvex minimization by the use of some nonmonotone line search. In this paper, we present a new non-descent backtracking type line search and show that the PRP method has strongly global convergence property i.e. lim k→∞‖∇ fxk‖=0 and locally R-linear convergence rate for nonconvex optimization with the proposed line search by suitably choosing the initial stepsize. Some numerical results compared with existing descent type line searches are reported.

In this paper, a modified Polak–Ribi‘ere–Polyak MPRP conjugate gradient method for smooth unconstrained optimization is proposed. This method produces at each iteration a descent direction, and this property is independent of the line search adopted. Under standard assumptions, we prove that the MPRP method using strong Wolfe conditions is globally convergent. The results of computational experiments are reported and show the effectiveness of the proposed method.

The equilibrium problem (EP) can be formulated as an unconstrained minimization problem through the D-gap function. We present a descent type algorithm for solving EP based on the generalized D-gap function. We discuss the convergence properties of the proposed algorithm under suitable assumptions while supporting our approach with appropriate examples. We construct an error bound for the equilibrium problem interms of the generalized D-gap function which gives a significant modification to the error bound given by Konnov et al.

New algorithms for solving unconstrained optimization problems are presented based on the idea of combining two types of descent directions: the direction of anti-gradient and either the Newton or quasi-Newton directions. The use of latter directions allows one to improve the convergence rate. Global and superlinear convergence properties of these algorithms are established. Numerical experiments using some unconstrained test problems are reported. Also, the proposed algorithms are compared with some existing similar methods using results of experiments. This comparison demonstrates the efficiency of the proposed combined methods.

In this article we will give relation between ascent and descent of a Banach space operator T and its diagonal (i.e. its restriction A to an invariant subspace and the induced quotient mapping B). This result is then applied to describe Drazin invertibility of one of these three operators using Drazin invertibility of the other two operators. It is proved that the operator T is meromorphic if and only if A and B are meromorphic.

We characterize the ascent and descent directions for the modulus of a complex polynomial p(z) at an arbitrary point z0 in the complex plane. We prove that when p(z0) ≠ 0, the cones of ascent and d...

In this article we propose two algorithms for minimization of quasiconvex functions. The first one is of type subgradient. In the second one, we consider the steepest descent method with Armijo's rule. In both, we use elements from Plastria's lower subdifferential. Under certain conditions, we prove that the sequence generated by these algorithms globally converges to a solution. We provide a counter-example showing that the choice of the minus gradient direction does not assure the global convergence of the descent method to a solution. This counter-example is related to a mistake in the proof of the Theorem 3.1 of J.P. Dussault, [Convergence of implementable descent algorithms for unconstrained optimization (technical note), J. Optim. Theory Appl. 104 (2000), pp. 739–745]. We also point out the mistake in the proof of that theorem.

It was earlier demonstrated, by the so-called main (or simplest) problem of the Calculus of Variations, that the Theory of Exact Penalties allows one not only to derive fundamental results of the Calculus of Variations but also to construct new direct numerical methods for solving variational problems based on the notions of subgradient and hypogradient of the exact penalty function (which is essentially nonsmooth even if all initial data are smooth). In this article Exact Penalties are used to solve isoperimetric problems of the Calculus of Variations. New direct numerical methods are described (e.g. the method of hypodifferential descent). Several numerical examples are discussed.

In this article, a new method is proposed for solving symmetric nonlinear equations, which can ensure that the search direction is descent for the norm function without carrying any line search technique. Under mild conditions, the global convergence of the given method is established. Numerical results show that the proposed method is effective for the given test problems.

We present an algorithm to locally minimize nonsmooth, nonconvex functions. In order to find descent directions, the notion of quasisecants, introduced in this paper, is applied. We prove that the algorithm converges to Clarke stationary points. Numerical results are presented demonstrating the applicability of the proposed algorithm to a wide variety of nonsmooth, nonconvex optimization problems. We also compare the proposed algorithm with the bundle method using numerical results.

A globally convergent algorithm for equilibrium problems with differentiable bifunctions is proposed. The algorithm is based on descent directions of a suitable family of gap functions. The novelty of the approach is that assumptions which guarantee that the stationary points of the gap functions are global optima are not required.

An iterative strategy for the reconstruction of objects buried in a medium and the identification of their material parameters is analysed. The algorithm alternates guesses of the domains using topological derivatives with corrections of the parameters obtained by descent techniques. Numerical experiments in geometries with multiple scatterers show that our scheme predicts the number, location and shape of objects, together with their physical parameters, with reasonable accuracy in a few steps.

This paper studies solutions of some piecewise-linear difference equations. In two particular cases, a descent argument is used to show that all solutions are periodic with either prime period 3(2 k − 1) or 6(2 k − 1) for some k ≥ 1. The existence of solutions with such periods is also considered.

For solving monotone variational inequalities with separate structures, Ye and Yuan [A descent method for stuctured monotone variational inequalities, Optim. Methods Softw. 22 (2007), 329–338] used the iterates generated by the well-known alternating directions method to design a descent direction, and thus presented a contraction method. This article continues on this study. By observing an improved descent direction and, selecting the corresponding optimal step sizes, a new contraction method is presented. In addition to proving the algorithm's convergence under mild assumptions, we compare the improved contraction method to Ye and Yuan's method (which is generalized) and achieve the superiority of the new method in theoretical senses.

This article presents a descent method for solving monotone variational inequalities with separate structures. The descent direction is derived from the well-known alternating directions method. The optimal step size along the descent direction also improves the efficiency of the new method. Some numerical results demonstrate that the new method is effective in practice.

In this article we present a systematic study of the reflexivity properties of homologically finite complexes with respect to semidualizing complexes in the setting of nonlocal rings. One primary focus is the descent of these properties over ring homomorphisms of finite flat dimension, presented in terms of inequalities between generalized G-dimensions. Most of these results are new even when the ring homomorphism is local. The main tool for these analyses is a nonlocal version of the amplitude inequality of Iversen, Foxby, and Iyengar. We provide numerous examples demonstrating the need for certain hypotheses and the strictness of many inequalities.

The notions of exhaustive families of upper convex and lower concave approximations (in the sense of Pschenichnyi) were introduced by Rubinov. For some classes of nonsmooth functions, these tools appeared to be very productive and constructive (e.g., in the case of quasidifferentiable functions). Dual tools – the upper exhauster and the lower exhauster – can be employed to describe optimality conditions and to find directions of the steepest ascent and descent. If a proper exhauster is known (for minimality conditions we need an upper exhauster, while for maximality ones a lower exhauster is required), the above problems are reduced to the problems of finding the nearest points to convex sets. If we study, e.g., the minimization problem and a lower exhauster is available, it is required to convert it into an upper one. In the present article it is shown how to use a lower exhauster to get conditions for a minimum without converting the lower exhauster into an upper one.

Assume that ϕ:(R, ± 𝔪) → (S, ± 𝔫) is a local flat homomorphism between commutative Noetherian local rings R and S. Let M be a finitely generated R-module. We investigate the ascent and descent of sequentially Cohen-Macaulay properties between the R-module M and the S-module M ⊗ R S.

International audience

We describe a two-semester numerical methods course that serves as a research experience for undergraduates without requiring external funding or modification of current curriculum. The first semester introduces traditional material and builds a proper set of tools that the students use in the second semester to approach a more research oriented problem. Our vehicle is an engineering problem associated with the hydro-dynamics of keel design which is used to introduce students to constrained optimization via a variation of the traditional isoperimetric problem of finding the curve with fixed end-points, fixed perimeter, and maximum area.

Given a Galois embedding problem H → G = Gal(L/F) with kernel of order two and F Hilbertian, we consider how obstructions OK to subgroup embedding problems H 0 → G 0 = Gal(L/K) for [K : F] = 2 descend to the obstruction OF to the original embedding problem, up to Br2(K/F). In particular, to such an obstruction OK we associate a tower of Z/2Z-embedding problems and prove that the contribution of OK to OF is given by the obstruction to the last embedding problem in the tower. We show that such an association in fact holds generally for central, Brauer Z/pZ-problems. When OK is the class of a quaternion algebra, we give an explicit representation of OF up to Br2(K/F). As a consequence, we represent in terms of quaternion algebras over F the obstructions to Z/2Z × Z/2Z-embedding problems not previously determined over F.

Recall from [D1] that the commutative domain R is a going down domain (or R is GD) if for any extension domain T of R, R ⊆ T satisfies going down, i.e., if Q ∈ Spec T and p ∈ Spec R with p ⊆ Q ∩ R,...

We prove two lemmata about Schubert calculus on generalized flag manifolds G/B, and in the case of the ordinary flag manifold GL_n/B we interpret them combinatorially in terms of descents, and geometrically in terms of missing subspaces. One of them gives a symmetry of Schubert calculus that we christen_descent-cycling_. Computer experiment shows that these lemmata suffice to determine all of GL_n Schubert calculus through n=5, and 99.97%+ at n=6. We use them to give a quick proof of Monk's rule. The lemmata also hold in equivariant (``double'') Schubert calculus for Kac-Moody groups G., 10 pages, 2 figures, see also http://www.math.berkeley.edu/~allenk/java/DCApplet.html

Global convergence theorems for a class of descent methods for unconstrained optimization problems in normed spaces, using multidirectional search, are proved. Exact and inexact search are considered and the results allow to define a globally convergent algorithm for an unconstrained optimal control problem which operates, at each step, on discrete approximations of the original continuous problem.

We show that in the category Kelley of Hausdorff k-spaces a map is exponentiable if and only if it is open and that any open surjection is an effective descent morphism.

The problem of time-continuous meteorological data assimilation is addressed. An optimal control method is proposed and studied for a simple well-posed 2D atmospheric model and continuous in time and space observations. An existence result for the optimal control problem is given. Then the first order necessary conditions of optimality are explicited using a forced version of the linearized adjoint model; allowing the use of a classical descent method to solve the problem. The reliability of the method is shown on some numerical tests. Finally, the existence and characterization results of an optimal control are extended to the case of observations distributed in time.

In this paper, we consider the optimization method for monotone variational inequality probleln on polyhedral sets. First, we consider the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed and s desirable properties of the merit function are obtained. Under certain assumptions, we show tthat any stationary point of the merit function is a solution of the original problem. A descent metlpod for the optimization problem is proposed and the global convergence of the method is shown

We consider a variational inequality problem, where the cost mapping is the sum of a single-valued mapping and the subdifferential mapping of a convex function. For this problem we introduce a new class of equivalent optimi7ation formulations; based on them, we also provide the first convergence analysis of descent algorithms for the problem. The optimization formulations constitute generalizations of those presented by Auchmuty [Auc89]. and the descent algorithms are likewise generalizations of those of Fukushima [Fuk92], Larsson and Patriksson [LaP94] and several others, for variational inequality problems with single-valued cost mappings

In a compa nion paper (A new algorithm to solve calculus of variations problems using Wolfe's convergence theory. Part I: Theory and algorithm), a new a lgorithm with the proper ties of guaranteed finite term in ati on and global co nvergence was p roposed to so lve calculus of variations problems. A detailed impleme ntation of the algori thm is presented here . Th e impl ementat ion proposes a variable dimension method and incorp orat es a stable procedure to solve for the descent search direction . Numerical examples are given to illustrate the algorithm

Necessary minimum conditions on an equality-type constraint and various steepest decent (or descent) directions are studied with respect to the dependence on the special form of a quasidifferential. It turns out that all of them are independent of the different equivalent quasidifferentials