Search

In this paper, we consider a partial sparse and partial group sparse optimization problem, where the loss function is a continuously differentiable function (possibly nonconvex), and the penalty term consists of two parts associated with sparsity and group sparsity. The first part is the ℓ 0 norm of x , the second part is the ℓ 2 , 0 norm of y , i.e., λ 1 ‖ x ‖ 0 + λ 2 ‖ y ‖ 2 , 0 , where (x , y) ∈ R n + m is the decision variable. We give a continuous relaxation model of the above original problem, where the two parts of the penalty term are relaxed by Capped- ℓ 1 of x and group Capped- ℓ 1 of y respectively. Firstly, we define two kinds of first-order stationary points of the relaxation model. Based on the lower bound property of d-stationary points of the relaxation model, we establish the equivalence of solutions of the original problem and the relaxation model, which provides a theoretical basis for solving the original problem via solving the relaxation problem. Secondly, we propose an alternating proximal gradient (APG) algorithm to solve the relaxation model, and prove that the whole sequence of the APG algorithm converges to a critical point under some mild conditions. Finally, numerical experiments on simulated data and multichannel image as well as comparison with some state-of-art algorithms are presented to illustrate the effectiveness and robustness of the proposed algorithm for partial sparse and partial group sparse optimization problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the matrix optimization problem with group sparsity and nonnegativity constraints. We analyze the optimality conditions and develop two matrix-based improved iterative hard thresholding algorithms for the problem, using the projected gradient method with the Armijo-type stepsize rule and the fixed stepsize, respectively. We then prove that the whole sequence generated by each of the proposed algorithm converges to a local minimizer of the optimization problem and establish the linear convergence rates as well as the iteration complexity results under mild conditions. Finally, numerical results are reported to demonstrate the usefulness of this type of problem and the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Abstract This study evaluated the effects of broiler age (A) and levels of replacement (L) of control diet (CD) on the utilization of energy and nutrients of whole corn germ. 720 one-day-old broilers (b) were allocated at completely randomized design to six treatments and six replicates, in three assays: pre-starter (1-8 days, 10 b/cage), starter (15-22 days, 6 b/cage), and grower (28-35 days, 4 b/cage) phases. The treatments were: CD and four test diets (L): 100, 150, 200, 250, or 300 g kg-1 of the CD replaced by WCG levels. The data were adjusted to the response surface model. The stationary points for apparent energy metabolizable (AME) and AME corrected for nitrogen balance (AMEn) were: 4173 and 3591 kcal kg-1, respectively, and coefficients of gross energy (AMCGE), crude protein (AMCCP), dry matter (AMCDM), and ether extract (AMCEE) were: 49.3, 40.4, 72.6, and 61.3%, respectively; and Ileal digestibility coefficient of crude protein (IDCCP), dry matter (IDCDM), digestibility crude protein values (DCP), and digestibility dry matter value (DDM) were: 78.0, 57.96, 8.50, and 56.17%, respectively. The EP for AMEn was at 18 days of age, 28 g kg-1 WCG. There was a correlation between A and L on digestibility and metabolisability of nutrient’s WCG.

In this paper, we consider a class of possibly non-convex and non-smooth optimization problems arising in many contemporary applications such as machine learning, variable selection and image processing. To solve this class of problems, we propose a proximal gradient method with extrapolation and line search (PGels). This method is developed based on a special potential function and successfully incorporates both extrapolation and non-monotone line search, which are two simple and efficient acceleration techniques for the proximal gradient method. Thanks to the non-monotone line search, this method allows more flexibility in choosing the extrapolation parameters and updates them adaptively at each iteration if a certain criterion is not satisfied. Moreover, with proper choices of parameters, our PGels reduces to many existing algorithms. We also show that, under some mild conditions, our line search criterion is well defined and any cluster point of the sequence generated by the PGels is a stationary point of our problem. In addition, by making assumptions on the Kurdyka–Łojasiewicz exponent of the objective in our problem, we further analyze the local convergence rate of two special cases of the PGels, including the widely used non-monotone proximal gradient method as one case. Finally, we conduct some preliminary numerical experiments for solving the ℓ 1 regularized logistic regression problem and the ℓ 1 - 2 regularized least squares problem. The obtained numerical results show the promising performance of the PGels and validate the potential advantage of combining two acceleration techniques. [ABSTRACT FROM AUTHOR]

Despite its popularity in the reinforcement learning community, a provably convergent policy gradient method for continuous space-time control problems with nonlinear state dynamics has been elusive. This paper proposes proximal gradient algorithms for feedback controls of finite-time horizon stochastic control problems. The state dynamics are nonlinear diffusions with control-affine drift, and the cost functions are nonconvex in the state and nonsmooth in the control. The system noise can degenerate, which allows for deterministic control problems as special cases. We prove under suitable conditions that the algorithm converges linearly to a stationary point of the control problem and is stable with respect to policy updates by approximate gradient steps. The convergence result justifies the recent reinforcement learning heuristics that adding entropy regularization or a fictitious discount factor to the optimization objective accelerates the convergence of policy gradient methods. The proof exploits careful regularity estimates of backward stochastic differential equations. [ABSTRACT FROM AUTHOR]

We consider a class of sparsity-inducing optimization problems whose constraint set is regularizer-compatible, in the sense that, the constraint set becomes easy-to-project-onto after a coordinate transformation induced by the sparsity-inducing regularizer. Our model is general enough to cover, as special cases, the ordered LASSO model in Tibshirani and Suo (Technometrics 58:415–423, 2016) and its variants with some commonly used nonconvex sparsity-inducing regularizers. The presence of both the sparsity-inducing regularizer and the constraint set poses challenges on the design of efficient algorithms. In this paper, by exploiting absolute-value symmetry and other properties in the sparsity-inducing regularizer, we propose a new algorithm, called the doubly majorized algorithm (DMA), for this class of problems. The DMA makes use of projections onto the constraint set after the coordinate transformation in each iteration, and hence can be performed efficiently. Without invoking any commonly used constraint qualification conditions such as those based on horizon subdifferentials, we show that any accumulation point of the sequence generated by DMA is a so-called ψ opt -stationary point, a new notion of stationarity we define as inspired by the notion of L-stationarity in Beck and Eldar (SIAM J Optim 23:1480–1509, 2013) and Beck and Hallak (Math Oper Res 41:196–223, 2016). We also show that any global minimizer of our model has to be a ψ opt -stationary point, again without imposing any constraint qualification conditions. Finally, we illustrate numerically the performance of DMA on solving variants of ordered LASSO with nonconvex regularizers. [ABSTRACT FROM AUTHOR]

In multivariate regression analysis, a coefficient matrix is used to relate multiple response variables to regressor variables in a noisy linear system from given data. Optimization is a natural approach to find such coefficient matrix with few nonzero rows. However, the relationship of most group sparse optimization models with cardinality penalty or cardinality constraints is not clear. In this paper, we give a comprehensive description of the relationship between three widely used group sparse optimization problems with cardinality terms: (i) the number of nonzero rows is minimized subject to an error tolerance for regression; (ii) the error for regression is minimized subject to a row cardinality constraint; (iii) the sum of the number of nonzero rows and error for regression is minimized. The first two problems have convex constraints and cardinality constraints respectively, while the third one is an unconstrained optimization problem with a cardinality penalty. We provide sufficient conditions under which the three optimization problems have the same global minimizers. Moreover, we analyze the relationship of stationary points and local minimizers of the three problems. Finally, we use two examples to illustrate our theoretical results for finding solutions of constrained optimization problems involving cardinality terms by unconstrained optimization problems with penalty functions. [ABSTRACT FROM AUTHOR]

Visitor crowding issues caused by over-tourism at World Heritage Sites (WHS) are increasingly prominent. However, related studies and management give less consideration to the uneven tourist spatial-temporal distribution within micro-attractions, ignoring the detrimental impacts of local crowding on WHS' sustainable development. Aimed at exploring local crowding within heritage attractions from a perspective of time-space, this study used video-based computer vision technology to analyze tourist spatial-temporal distribution, density, and carrying capacity at stationary points (spatial locations where visitors stay and gather), taking a case study of the Master-of-Nets Garden, a World Cultural Heritage Site. The results indicated that although the total visitor number was below the attraction's carrying capacity, visitors repeatedly and quickly crowded at stationary points and formed high-density clusters, exceeding stationary points' space capacity and causing spatial local crowding. When visitors' dwell time was 5, 12, and 20 seconds, the crowding index K of stationary points was 3.4∼108.5, 1.7∼42.9, and 0.5∼33, indicating the spatial local crowding effect was negatively correlated with the length of visitors' dwell time at stationary points. Our contribution provides an accurate understanding of local crowding in WHS by highlighting the impact of tourist spatial-temporal distribution on carrying capacity and spatial crowding and suggests visitor management recommendations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a conditional gradient method for solving constrained vector optimization problems with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. When the partial order under consideration is the one induced by the non-negative orthant, we regain the method for multiobjective optimization recently proposed by Assunção et al. (Comput Optim Appl 78(3):741–768, 2021). In our method, the construction of the auxiliary subproblem is based on the well-known oriented distance function. Three different types of step size strategies (Armijo, adaptative and nonmonotone) are considered. Without convexity assumption related to the objective function, we obtain the stationarity of accumulation points of the sequences produced by the proposed method equipped with the Armijo or the nonmonotone step size rule. To obtain the convergence result of the method with the adaptative step size strategy, we introduce a useful cone convexity condition which allows us to circumvent the intricate question of the Lipschitz continuity of Jocabian for the objective function. This condition helps us to generalize the classical descent lemma to the vector optimization case. Under convexity assumption for the objective function, it is proved that all accumulation points of any generated sequences obtained by our method are weakly efficient solutions. Numerical experiments illustrating the practical behavior of the methods are presented. [ABSTRACT FROM AUTHOR]

With the development of science and technology, we can get many groups of data for the same object. There is a certain relationship with each other or structure between these data or within the data. To characterize the structure of the data in different datasets, in this paper, we propose a multiple sparse constraint problem (MSCP) to process the problem with multiblock sparse structure. We give three types of stationary points and present the relationships among the three types of stationary points and the global/local minimizers. Then we design a gradient projection Newton algorithm, which is proven to enjoy the global and quadratic convergence property. Finally, some numerical experiments of different examples illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The crop leaf color is tightly connected with its meteorological environment. Color gradation skewness-distribution (CGSD) parameters can describe the information of leaf color more accurately, systematically, and comprehensively from five dimensions. We took photographs of pepper growing in the greenhouse at a fixed time every day and observed the meteorological factors. The results showed that the CGSD parameters were significantly correlated with meteorological factors, especially with the accumulated temperature, which showed the strongest correlation. Since the relationship between canopy leaf color and accumulated temperature is nonlinear, the piecewise inversion models were constructed by taking the stationary point of the high-order response model of Gskewness to accumulated temperature as the point of demarcation. The rate of outliers had decreased by 57.72%; moreover, the overall inversion accuracy had increased by 3.31% compared with the linear model directly constructed by the stepwise regression. It was observed that the pepper in the greenhouse had a different response to the same meteorological environmental stimulus before and after the stationary point. This study will provide a new method for constructing crop growth models in future research. [ABSTRACT FROM AUTHOR]

We show that the incenter of an isosceles triangle is a stable equilibrium point of the electrostatic potential of certain point charges placed at its vertices. To this end, explicit formulas for these charges are given and the hessian of their electrostatic potential is computed. The behavior of this hessian in a family of triangles with the given inscribed and circumscribed circles is investigated and its extremal values are computed. As an application, we prove that each point in the unit disc is a stable equilibrium point of a certain triple of point charges on its boundary, which yields an explicit scenario of robust electrostatic control in Euclidean discs. In conclusion, several examples are presented which show that analogous problems are interesting and feasible for convex quadrilaterals. [ABSTRACT FROM AUTHOR]

In this paper, augmented Levin methods are proposed for the computation of oscillatory integrals with stationary points and an algebraically or logarithmically singular kernel. Different from the conventional Levin method, to overcome the difficulties caused by singular and stationary points, the original Levin ordinary differential equation (Levin-ODE) is converted into an augmented ODE system, which can be fast and stably implemented with a cost of O(n\log n) by applying sparse and fast spectral methods together with the truncated singular value decomposition. The established asymptotics and convergence show that these schemes become more accurate as the frequency increases and are super-algebraically convergent. The effectiveness and accuracy were tested by numerical examples, showing perfect coincidence with the estimates. [ABSTRACT FROM AUTHOR]

In the present article, by applying our 2013 Metatheorem and the Brøndsted-Jachymski Principle, we obtain various forms generalizations of Zorn's Lemma and their applications. Such examples are our version of the Zermelo fixed point theorem, equivalent formulations of the Caristi fixed point theorem, and Jachymski's 2003 theorem on equivalent conditions when fixed point sets are same to periodic point sets. [ABSTRACT FROM AUTHOR]

The modified Filon–Clenshaw–Curtis rules, proposed earlier by the author, are combined with the (double) exponential transformations in such a way that (1) the necessity of computing the inverse of the oscillator function is released, (2) possible inaccuracy due to rounding error, when the amplitude function has endpoint singularities, is treated, (3) difficulties raised by the stationary points of the oscillator function are treated, and (4) all the benefits of the original Filon–Clenshaw–Curtis rules are preserved. We also carry out some numerical experiments, which illustrate the efficiency of the proposed algorithms while earlier methods based on Filon–Clenshaw–Curtis rules result in poor approximations or even fail. [ABSTRACT FROM AUTHOR]

The crop leaf color is tightly connected with its meteorological environment. Color gradation skewness-distribution (CGSD) parameters can describe the information of leaf color more accurately, systematically, and comprehensively from five dimensions. We took photographs of pepper growing in the greenhouse at a fixed time every day and observed the meteorological factors. The results showed that the CGSD parameters were significantly correlated with meteorological factors, especially with the accumulated temperature, which showed the strongest correlation. Since the relationship between canopy leaf color and accumulated temperature is nonlinear, the piecewise inversion models were constructed by taking the stationary point of the high-order response model of Gskewness to accumulated temperature as the point of demarcation. The rate of outliers had decreased by 57.72%; moreover, the overall inversion accuracy had increased by 3.31% compared with the linear model directly constructed by the stepwise regression. It was observed that the pepper in the greenhouse had a different response to the same meteorological environmental stimulus before and after the stationary point. This study will provide a new method for constructing crop growth models in future research.

In this paper, we study a first-order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems and introduce two concepts of critical points. Furthermore, we propose a descent method and provide a convergence result to points satisfying the optimality conditions previously derived. Some numerical examples illustrating the performance of the method are also discussed. This paper is a modified and polished version of Chapter 5 in the dissertation by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms, Martin-Luther-Universität Halle-Wittenberg, 2020). [ABSTRACT FROM AUTHOR]

In this paper, we are committed to the study of generalized difference gap (D-gap) functions and error bounds for hemivariational inequalities in Hilbert spaces. By introducing a new gap function, we define a generalized D-gap function for the hemivariational inequality considered, for which we investigate the local Lipschitz continuity and coercivity. Then, some error bound results for the generalized D-gap function are established and the relationship between the solution to the hemivariational inequality and the stationary point of the generalized D-gap function is discussed. Finally, we construct a descent algorithm for solving the hemivariational inequality based on the generalized D-gap function and further prove a convergence result for the algorithm. [ABSTRACT FROM AUTHOR]

In the present paper a plane round double-hinged arch under the potential dead load is investigated. To describe the stress-strain state and the equilibrium stability the geometrically exact theory is used. According to this theory every point of the bar has two translational degrees of freedom and one rotational, which is independent from the previous two. To solve the problem no displacements are simplified and all the stiffnesses are used: axial, shear and bending. Exact nonlinear differential equations are found for the static problem. A variational definition for the problem is defined as finding a stationary point of Lagrange functional. The match of the differential and variational formulations is shown. Exact stability equations accounting non-linear geometric deformations in pre-buckling state were worked out. The problem of the equilibrium stability of the round arch under the potential dead load was solved using the obtained equations regarding all the bar’s stiffnesses. The characteristic transcendental equation and its asymptotic solution as simple formulas, suitable for practical application, were worked out. The comparison of described solution which regards all the bar’s stiffnesses and classical solution, based on bending stiffness, was made.

In this paper, we derive the sufficient condition for global optimality for a nonsmooth mathematical program with equilibrium constraints involving generalized invexity. We formulate the Wolfe and Mond-Weir type dual models for the problem using convexificators. We establish weak and strong duality theorems to relate the mathematical program with equilibrium constraints and the dual models in the framework of convexificators.

A broadband SiGe power amplifier (PA) with double gain peaks for 5G applications is presented. The differential structure is widely adopted in millimeter-wave circuit design to decrease the impact of the source parasitic inductance that is induced by the bonding wires. Since differential PA results in a higher impedance transformation ratio of the matching network, differential PAs generally have narrower bandwidth than single-ended PAs. Through the bandwidth analysis of the matching network, the transformer used for interstage matching is elaborately designed to ease gain fluctuations. By controlling the stationary points of impedance transformation, the maximum available gain characteristic of a transistor, which decreases with increasing frequency, can be compensated. The gain curve of PA, thus, has double gain peaks within operational frequencies to make its frequency responses as flat as possible. The proposed PA for the frequency band from 27 to 37 GHz is fabricated in a standard 130-nm SiGe BiCMOS process, which occupies 0.36 mm2. Under the 2.5-V voltage supply, the SiGe PA exhibited a power gain of 33.5 dB and output power of 21.3 dBm with 24% power-added efficiency at 31 GHz. It achieves a 34% 3-dB small-signal S21 bandwidth from 27.3 to 38.5 GHz. [ABSTRACT FROM AUTHOR]

We consider a controlled system of Lotka–Volterra differential equations that describes the evolution of two interrelated populations of predators and prey. The system contains two control variables, which are chosen so as to minimize the time of transition to a stationary point. The control functions and the corresponding trajectories of motion in the state space are constructed and their optimality is substantiated. [ABSTRACT FROM AUTHOR]

Copyright of Cybernetics & Systems Analysis / Kibernetiki i Sistemnyj Analiz is the property of V.M. Glushkov Institute of Cybernetics of NAS of Ukraine and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Algorithms based on the hard thresholding principle have been well studied with sounding theoretical guarantees in the compressed sensing and more general sparsity-constrained optimization. It is widely observed in existing empirical studies that when a restricted Newton step was used (as the debiasing step), the hard-thresholding algorithms tend to meet halting conditions in a signicantly low number of iterations and are very efficient. Hence, the thus obtained Newton hard-thresholding algorithms call for stronger theoretical guarantees than for their simple hard-thresholding counterparts. This paper provides a theoretical justication for the use of the restricted Newton step. We build our theory and algorithm, Newton Hard-Thresholding Pursuit (NHTP), for the sparsity-constrained optimization. Our main result shows that NHTP is quadratically convergent under the standard assumption of restricted strong convexity and smoothness. We also establish its global convergence to a stationary point under a weaker assumption. In the special case of the compressive sensing, NHTP effectively reduces to some of the existing hard-thresholding algorithms with a Newton step. Consequently, our fast convergence result justies why those algorithms perform better than without the Newton step. The efficiency of NHTP was demonstrated on both synthetic and real data in compressed sensing and sparse logistic regression. [ABSTRACT FROM AUTHOR]

In this paper, we consider a single-cell downlink multi-user multi-input multi-output (MU-MIMO) system in which multiple eavesdroppers coexist. The eavesdroppers overhear confidential information sent from a base station (BS). In particular, we assume a scenario of colluding eavesdroppers in which they cooperate with each other to decode the information. We develop a secure linear precoding method that maximizes the sum secrecy spectral efficiency. To this end, we formulate an optimization problem and propose a computationally-efficient algorithm that achieves a sub-optimal stationary point of the problem. Simulation results validate the proposed algorithm. The proposed method possesses several benefits: $i)$ it is applicable in general systems, for example, an arbitrary number of eavesdroppers and confidential information set, $ii)$ the associated complexity is relatively low, and $iii)$ it is also applicable when only channel covariance of the eavesdroppers is available. [ABSTRACT FROM AUTHOR]

A class of invex nonlinear complementarity (NCP) functions is introduced. The existence of such a function is illustrated by citing nontrivial examples. Some interesting properties of invex NCP functions have been developed. Quasi-invex NCP functions and their properties are also presented. [ABSTRACT FROM AUTHOR]

A novel ray-tracing algorithm to cope with the phase errors due to incorrect ray path computations in ray-launching approaches is presented. The algorithm utilizes bidirectional ray-tracing to collect information about wavefronts incident on an interaction surface and yields considerable improvements in accuracy compared to conventional unidirectional ray-tracing. The points, where exact ray paths intersect with the surface, are obtained according to the Fermat principle of least time. If the interaction surface is aligned with diffraction edges, the corresponding critical points of the second kind can also be retrieved and complicated diffraction treatments by shooting diffracted rays on Keller cones can be avoided. Thus, a substantial reduction in the number of rays can be achieved. Furthermore, a typical problem encountered in traditional ray-tracing due to the reception sphere mechanism, i.e., incorrect ray contributions, can mostly be evaded. Numerical results demonstrate the capabilities of the new algorithm and its advantages against traditional techniques. [ABSTRACT FROM AUTHOR]

We guarantee the existence of stationary points for semicompact lower semicontinuous multifunctions in metric spaces. We also show that the semicompactness condition can be replaced by some other properties. Several examples illustrating the main results are also given. [ABSTRACT FROM AUTHOR]

This paper is concerned with the computation of perfect stationary point, which is a strict refinement of stationary point. A differentiable homotopy method is developed for finding perfect stationary points of continuous functions on convex polytopes. We constitute an artificial problem by introducing a continuously differentiable function of an extra variable. With the optimality conditions of this problem and a fixed point argument, a differentiable homotopy mapping is constructed. As the extra variable becomes close to zero, the homotopy path naturally provides a sequence of closely approximate stationary points on perturbed polytopes, and converges to a perfect stationary point on the original polytope. Numerical experiments are implemented to further illustrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this paper, we consider matrix optimization with the variable as a matrix that is constrained into a low-rank spectral set, where the low-rank spectral set is the intersection of a low-rank set and a spectral set. Three typical spectral sets are considered, yielding three low-rank spectral sets. For each low-rank spectral set, we first calculate the projection of a given point onto this set and the formula of its normal cone, based on which the induced stationary points of matrix optimization over low-rank spectral sets are then investigated. Finally, we reveal the relationship between each stationary point and each local/global minimizer. [ABSTRACT FROM AUTHOR]

We propose a novel smoothing active set method for linearly constrained nonLipschitz nonconvex problems. At each step of the proposed method, we approximate the objective function by a smooth function with a fixed smoothing parameter and employ a new active set method for minimizing the smooth function over the original feasible set, until a special updating rule for the smoothing parameter meets. The updating rule is always satisfied within a finite number of iterations since the new active set method for smooth problems proposed in this paper forces at least one subsequence of projected gradients to zero. Any accumulation point of the smoothing active set method is a stationary point associated with the smoothing function used in the method, which is necessary for local optimality of the original problem. And any accumulation point for the \ell 2 - \ell p (0 < p < 1) sparse optimization model is a limiting stationary point, which is a local minimizer under a certain second-order condition. Numerical experiments demonstrate the efficiency and effectiveness of our smoothing active set method for hyperspectral unmixing on a 3 dimensional image cube of large size. [ABSTRACT FROM AUTHOR]

The spiral optimization (SPO) algorithm proposed by Tamura and Yasuda is a relatively novel and simple search concept inspired by natural spiral phenomena. This algorithm searches continuous space using no gradient and only spiral trajectories composed of spiral vectors generated by deterministic spiral models. The primary purpose of this paper is to propose conditions and settings that mathematically ensure the convergence of the SPO algorithm to a stationary point. The conditions relating to the sizes and directions of the spiral vectors and the initial search points are based on direct search theory and recent SPO algorithm theories. The presented convergence was numerically verified using test functions with different properties. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of structured nonsmooth optimization problem in which the first component of the objective is a smooth function while the second component is the sum of one-dimensional nonsmooth functions. We first verify that every minimizer of this problem is a solution of an equation h (x) = 0 , where h is continuous but not differentiable, and moreover - h (x) is a descent direction of the objective at x ∈ R n if h (x) ≠ 0 . Then by using - h (x) as a search direction, we propose a residual-based algorithm for solving this problem. Under proper conditions, we verify that any accumulation point of the sequence of iterates generated by our algorithm is a first-order stationary point of the problem. Additionally, we prove that the worst-case iteration-complexity for finding an ϵ first-order stationary point is O (ϵ - 2) . Numerical results have shown the efficiency of this algorithm. [ABSTRACT FROM AUTHOR]