Search

Trends over the past two decades indicate that much of the performance gains of commercial optimization solvers is due to improvements in x86 hardware. To continue making progress, it is critical to consider alternative/specialized massively parallel computing architectures. In this work, we detail the development of an open-source source code transformation approach built using Symbolics.jl to construct McCormick-based relaxations of functions that enables their effective parallelized evaluation. We then apply this approach in a novel parallelized branch-and-bound routine that offloads lower- and upper-bounding problems to a GPU. The effectiveness of this new approach is demonstrated on three nonconvex problems of interest, where it yields convergence time improvements of 22–118x compared to an equivalent serial CPU implementation and in two cases outperforms vanilla branch-and-bound versions of existing state-of-the-art solvers that use tighter bounding techniques. This work exemplifies how deterministic global optimizers using alternative hardware architectures can compete with—or eventually outclass—even the most powerful serial CPU implementations, and to the best of the authors' knowledge, represents the first successful demonstration of deterministic global optimization using a GPU. [ABSTRACT FROM AUTHOR]

This paper studies the convergence of the proximal point method for quasiconvex functions in finite dimensional complete Riemannian manifolds. We prove initially that, in the general case, when the objective function is proper and lower semicontinuous, each accumulation point of the sequence generated by the method, if it exists, is a limiting critical point of the function. Then, under the assumptions that the sectional curvature of the manifold is bounded above by some non negative constant and the objective function is quasiconvex we analyze two cases. When the constant is zero, the global convergence of the algorithm to a limiting critical point is assured and if it is positive, we prove the local convergence for a class of quasiconvex functions, which includes Lipschitz functions. [ABSTRACT FROM AUTHOR]

In this study, we investigate a class of linear multiplicative programs with positive exponents. By introducing p additional variables, the original problem is reformulated into an equivalent problem (EP) within the output space. Subsequently, a novel global optimization algorithm is introduced to tackle EP. The algorithm primarily leverages two key techniques. One is the outer approximation technique, which tightens the relaxed feasible region of EP and improves the upper bound by carefully examining suitable feasible points. The other is the branch and bound technique to guarantee the global optimality of the solution. Numerical results confirm the effectiveness and practicality of the proposed algorithm, thereby underscoring its potential for real-world applications. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified Tseng's extragradient method with a new step-length rule to solve pseudo-monotone variational inequalities in real Hilbert spaces. Under suitable conditions, the sequence generated by this algorithm strongly converges to the common elements of the solution set of pseudo-monotone variational inequality problems and the fixed point set of k-demicontractive mappings. Finally, we give some numerical experiments to illustrate the effectiveness of the proposed algorithm. The main results of this paper generalize and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, for solving a broad class of large-scale nonconvex and nonsmooth optimization problems, we propose a stochastic two-step inertial Bregman proximal alternating linearized minimization (STiBPALM) algorithm with variance-reduced stochastic gradient estimators. And we show that SAGA and SARAH are variance-reduced gradient estimators. Under expectation conditions with the Kurdyka–Łojasiewicz property and some suitable conditions on the parameters, we obtain that the sequence generated by the proposed algorithm converges to a critical point. And the general convergence rate is also provided. Numerical experiments on sparse nonnegative matrix factorization and blind image-deblurring are presented to demonstrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Tensor spectral p -norms are generalizations of matrix induced norms. Matrix induced norms are an important type of matrix norms, and tensor spectral p -norms are also important in applications. We discuss some basic properties of tensor spectral p -norms. We extend the submultiplicativity of the matrix spectral 2-norm to the tensor case, based on which we give a bound of the tensor spectral 2-norm and provide a fast method for computing spectral 2-norms of sum-of-squares tensors. To compute tensor spectral p -norms, we propose a higher order power method. Experiments show the high efficiency of the proposed methods and numerical results on spectral p -norms of random tensors are also given. [ABSTRACT FROM AUTHOR]

In this paper, we consider large-scale ranking problems where one is given a set of (possibly non-redundant) pairwise comparisons and the underlying ranking explained by those comparisons is desired. We show that stochastic gradient descent approaches can be leveraged to offer convergence to a solution that reveals the underlying ranking while requiring low-memory operations. We introduce several variations of this approach that offer a tradeoff in speed and convergence when the pairwise comparisons are noisy (i.e., some comparisons do not respect the underlying ranking). We prove theoretical results for convergence almost surely and study several regimes including those with full observations, partial observations, and noisy observations. Our empirical results give insights into the number of observations required as well as how much noise in those measurements can be tolerated. [ABSTRACT FROM AUTHOR]

We consider first-order methods with constant step size for minimizing locally Lipschitz coercive functions that are tame in an o-minimal structure on the real field. We prove that if the method is approximated by subgradient trajectories, then the iterates eventually remain in a neighborhood of a connected component of the set of critical points. Under suitable method-dependent regularity assumptions, this result applies to the subgradient method with momentum, the stochastic subgradient method with random reshuffling and momentum, and the random-permutations cyclic coordinate descent method. [ABSTRACT FROM AUTHOR]

We study a new variant of policy gradient method, named homotopic policy mirror descent (HPMD), for solving discounted, infinite horizon MDPs with finite state and action spaces. HPMD performs a mirror descent type policy update with an additional diminishing regularization term, and possesses several computational properties that seem to be new in the literature. When instantiated with Kullback–Leibler divergence, we establish the global linear convergence of HPMD applied to any MDP instance, for both the optimality gap, and a weighted distance to the set of optimal policies. We then unveil a phase transition, where both quantities exhibit local acceleration, and converge at a superlinear rate after the optimality gap falls below certain instance-dependent threshold. With local acceleration and diminishing regularization, we establish the first result among policy gradient methods on certifying and characterizing the limiting policy, by showing, with a non-asymptotic characterization, that the last-iterate policy converges to the unique optimal policy with the maximal entropy. We then extend all the aforementioned results to HPMD instantiated with a broad class of decomposable Bregman divergences, demonstrating the generality of the these computational properties. As a byproduct, we discover the finite-time exact convergence for some commonly used Bregman divergences, implying the continuing convergence of HPMD to the limiting policy even if the current policy is already optimal. Finally, we develop a stochastic version of HPMD and establish similar convergence properties. By exploiting the local acceleration, we show that when a generative model is available for policy evaluation, with a small enough ϵ 0 , for any target precision ϵ ≤ ϵ 0 , an ϵ -optimal policy can be learned with O ~ (| S | | A | / ϵ 0 2) samples with probability 1 - O (ϵ 0 1 / 3 ) . [ABSTRACT FROM AUTHOR]

We consider the multilinear polytope defined as the convex hull of the set of binary points z, satisfying a collection of equations of the form z e = ∏ v ∈ e z v for all e ∈ E . The complexity of the facial structure of the multilinear polytope is closely related to the acyclicity degree of the underlying hypergraph. We obtain a polynomial-size extended formulation for the multilinear polytope of β -acyclic hypergraphs, hence characterizing the acyclic hypergraphs for which such a formulation can be constructed. [ABSTRACT FROM AUTHOR]

In this work, we develop a new complexity metric for an important class of low-rank matrix optimization problems in both symmetric and asymmetric cases, where the metric aims to quantify the complexity of the nonconvex optimization landscape of each problem and the success of local search methods in solving the problem. The existing literature has focused on two recovery guarantees. The RIP constant is commonly used to characterize the complexity of matrix sensing problems. On the other hand, the incoherence and the sampling rate are used when analyzing matrix completion problems. The proposed complexity metric has the potential to generalize these two notions and also applies to a much larger class of problems. To mathematically study the properties of this metric, we focus on the rank-1 generalized matrix completion problem and illustrate the usefulness of the new complexity metric on three types of instances, namely, instances with the RIP condition, instances obeying the Bernoulli sampling model, and a synthetic example. We show that the complexity metric exhibits a consistent behavior in the three cases, even when other existing conditions fail to provide theoretical guarantees. These observations provide a strong implication that the new complexity metric has the potential to generalize various conditions of optimization complexity proposed for different applications. Furthermore, we establish theoretical results to provide sufficient conditions and necessary conditions for the existence of spurious solutions in terms of the proposed complexity metric. This contrasts with the RIP and incoherence conditions that fail to provide any necessary condition. [ABSTRACT FROM AUTHOR]

This article presents a self-adjustable branch-and-bound algorithm for globally solving a class of linear multiplicative programming problems (LMP). In this algorithm, a self-adjustable branching rule is introduced and it can continuously update the upper bound for the optimal value of LMP by selecting suitable branching point under certain conditions, which differs from the standard bisection rule. The proposed algorithm further integrates the linear relaxation program and the self-adjustable branching rule. The dependability and robustness of the proposed algorithm are demonstrated by establishing the global convergence. Furthermore, the computational complexity of the proposed algorithm is estimated. Finally, numerical results validate the effectiveness of the self-adjustable branching rule and demonstrate the feasibility of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We study T-semidefinite programming (SDP) relaxation for constrained polynomial optimization problems (POPs). T-SDP relaxation for unconstrained POPs was introduced by Zheng et al. (JGO 84:415–440, 2022). In this work, we propose a T-SDP relaxation for POPs with polynomial inequality constraints and show that the resulting T-SDP relaxation formulated with third-order tensors can be transformed into the standard SDP relaxation with block-diagonal structures. The convergence of the T-SDP relaxation to the optimal value of a given constrained POP is established under moderate assumptions as the relaxation level increases. Additionally, the feasibility and optimality of the T-SDP relaxation are discussed. Numerical results illustrate that the proposed T-SDP relaxation enhances numerical efficiency. [ABSTRACT FROM AUTHOR]

First-order optimization methods have long been established as effective methods for solving large-scale unconstrained optimization problems. Over time, these methods have been extended to address the task of finding zeros of single-valued monotone maps within the framework of finite-dimensional Hilbert spaces. Notably, Abubakar et al. (Comput Appl Math 39(2):129, 2020) recently introduced a projection method that incorporates a convex combination of two distinct positive spectral coefficients to find the zeros of a single-valued monotone map. Motivated by their work, this paper generalises the Abubakar et al. technique to a more general setting of infinite-dimensional Hilbert spaces. Under mild assumptions, we establish weak convergence of the generated sequence of iterates. Furthermore, the extended method does not involve computing the resolvent of the monotone operator. [ABSTRACT FROM AUTHOR]

We study several discretizations of a second order gradient-like system with damping. We first consider an explicit scheme with a linear damping in finite dimension. We prove that every solution converges if the nonlinearity satisfies a global Lojasiewicz inequality. Convergence rates are also established. In the case of a strong nonlinear damping, we prove convergence of every solution for a fully implicit scheme in the one-dimensional case, even if the nonlinearity does not satisfy a Lojasiewicz inequality. The optimality of the damping is also established. Numerical simulations illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We investigate how to solve smooth matrix optimization problems with general linear inequality constraints on the eigenvalues of a symmetric matrix. We present solution methods to obtain exact global minima for linear objective functions, i.e., F (X) = ⟨ C , X ⟩ , and perform exact projections onto the eigenvalue constraint set. Two first-order algorithms are developed to obtain first-order stationary points for general non-convex objective functions. Both methods are proven to converge sublinearly when the constraint set is convex. Numerical experiments demonstrate the applicability of both the model and the methods. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sequential M-stationarity conditions for a class of nonsmooth nonconvex general optimization problems. We introduce various types of such conditions and compare them with previously established conditions in smooth or convex cases. The application of the derived results is demonstrated in the context of nonsmooth sparsity-constrained optimization problems. Additionally, we devise a Lagrangian-type algorithm for a specific case of smooth sparsity problems. Several examples are presented throughout the paper to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, an approximate descent three-term derivative-free algorithm is developed for a large-scale system of nonlinear symmetric equations where the gradients and the difference of the gradients are computed approximately in order to avoid computing and storing the corresponding Jacobian matrices or their approximate matrices. The new method enjoys the sufficient descent property independent of the accuracy of line search strategies and the error bounds of these approximations are established. Under some mild conditions and a nonmonotone line search technique, the global and local convergence properties are established respectively. Numerical results indicate that the proposed algorithm outperforms the other similar ones available in the literature. [ABSTRACT FROM AUTHOR]

This paper investigates sparse optimization problems characterized by a sparse group structure, where element- and group-level sparsity are jointly taken into account. This particular optimization model has exhibited notable efficacy in tasks such as feature selection, parameter estimation, and the advancement of model interpretability. Central to our study is the scrutiny of the ℓ 0 and ℓ 2 , 0 norm regularization model, which, in comparison to alternative surrogate formulations, presents formidable computational challenges. We embark on our study by conducting the analysis of the optimality conditions of the sparse group optimization problem, leveraging the notion of a γ -stationary point, whose linkage to local and global minimizer is established. In a subsequent facet of our study, we develop a novel subspace Newton algorithm for sparse group ℓ 0 optimization problem and prove its global convergence property as well as local second-order convergence rate. Experimental results reveal the superlative performance of our algorithm in terms of both precision and computational expediency, thereby outperforming several state-of-the-art solvers. [ABSTRACT FROM AUTHOR]

Nonconvex minimax problems have attracted significant attention in machine learning, wireless communication and many other fields. In this paper, we propose an efficient approximation proximal gradient algorithm for solving a class of nonsmooth nonconvex-linear minimax problems with a nonconvex nonsmooth term, and the number of iteration to find an ε -stationary point is upper bounded by O (ε - 3) . Some numerical results on one-bit precoding problem in massive MIMO system and a distributed non-convex optimization problem demonstrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The article proposes an exact approach to finding the global solution of a nonconvex semivectorial bilevel optimization problem, where the objective functions at each level are pseudoconvex, and the constraints are quasiconvex. Due to its non-convexity, this problem is challenging, but it attracts more and more interest because of its practical applications. The algorithm is developed based on monotonic optimization combined with a recent neurodynamic approach, where the solution set of the lower-level problem is inner approximated by copolyblocks in outcome space. From that, the upper-level problem is solved using the branch-and-bound method. Finding the bounds is converted to pseudoconvex programming problems, which are solved using the neurodynamic method. The algorithm's convergence is proved, and computational experiments are implemented to demonstrate the accuracy of the proposed approach. [ABSTRACT FROM AUTHOR]

Without setting any condition on the parameter θ k of a four-term version of the classical one-parameter Hager-Zhang (HZ) method, this article proposes another HZ-type scheme for solving constrained monotone equations, where the condition for global convergence is satisfied for θ k ∈ [ 0 , + ∞) . This is an improvement from the former, its recent adaptive variant, where the global convergence condition holds for θ k ∈ (0 , + ∞) under certain defined condition, as well as other adaptations for systems of monotone equations, where the condition holds only when θ k ∈ (1 4 , + ∞) . By conducting singular value study of iteration matrix of the scheme, a choice of θ k restricted in the interval (0 , 1 4 ] is obtained to study its impact on the scheme. Moreover, the scheme converges globally and its effectiveness is shown by some numerical experiments and image de-blurring application. [ABSTRACT FROM AUTHOR]

Multidimensional scaling (MDS) is a popular tool for dimensionality reduction and data visualization. Given distances between data points and a target low-dimension, the MDS problem seeks to find a configuration of these points in the low-dimensional space, such that the inter-point distances are preserved as well as possible. We focus on the most common approach to formulate the MDS problem, known as stress minimization, which results in a challenging non-smooth and non-convex optimization problem. In this paper, we propose an inertial version of the well-known SMACOF Algorithm, which we call AI-SMACOF. This algorithm is proven to be globally convergent, and to the best of our knowledge this is the first result of this kind for algorithms aiming at solving the stress MDS minimization. In addition to the theoretical findings, numerical experiments provide another evidence for the superiority of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper presents modified memoryless quasi-Newton methods based on the spectral-scaling Broyden family on Riemannian manifolds. The method involves adding one parameter to the search direction of the memoryless self-scaling Broyden family on the manifold. Moreover, it uses a general map instead of vector transport. This idea has already been proposed within a general framework of Riemannian conjugate gradient methods where one can use vector transport, scaled vector transport, or an inverse retraction. We show that the search direction satisfies the sufficient descent condition under some assumptions on the parameters. In addition, we show global convergence of the proposed method under the Wolfe conditions. We numerically compare it with existing methods, including Riemannian conjugate gradient methods and the memoryless spectral-scaling Broyden family. The numerical results indicate that the proposed method with the BFGS formula is suitable for solving an off-diagonal cost function minimization problem on an oblique manifold. [ABSTRACT FROM AUTHOR]

Robust principal component analysis (RPCA) is devoted to tackling grossly corrupted datasets with noise. However, the performance of RPCA is usually circumscribed by the lack of efficiency of singular value decomposition (SVD), which rules out its potential applications to many large-scale real-world problems. In this paper, we develop a nonconvex optimization algorithm customized to SVD-free RPCA models. The proposed algorithm, which is built upon proximal alternating linearized minimization Bolte et al. [Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math Program. 2014;146(1–2):459–494], can reduce computational efforts by partially linearizing data fidelity and increase efficiency by leveraging inertial techniques. Under the Kurdyka-Łojasiewicz assumption on the objective function and some mild premises on stepsizes, the sequence produced by the proposed algorithm converges globally to a critical point of SVD-free RPCA models. Numerical simulations on synthetic and real datasets demonstrate the compelling performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We investigate the strong convergence properties of a Nesterov type algorithm with two Tikhonov regularization terms in connection to the minimization problem of a smooth convex function f. We show that the generated sequences converge strongly to the minimal norm element from argmin f . We also show fast convergence for the potential energies f (x n) - min f and f (y n) - min f , where (x n) , (y n) are the sequences generated by our algorithm. Further we obtain fast convergence to zero of the discrete velocity and some estimates concerning the value of the gradient of the objective function in the generated sequences. Via some numerical experiments we show that we need both Tikhonov regularization terms in our algorithm in order to obtain the strong convergence of the generated sequences to the minimum norm minimizer of our objective function. [ABSTRACT FROM AUTHOR]

Many problems in data science can be treated as recovering structural signals from a set of linear measurements, sometimes perturbed by dense noise or sparse corruptions. In this paper, we develop a unified framework of considering a nonsmooth formulation with sparse or low-rank constraint for meeting the challenges of mixed noises—bounded noise and sparse noise. We show that the nonsmooth formulations of the problems can be well solved by the projected subgradient methods at a rapid rate when initialized at any points. Consequently, nonsmooth loss functions ( ℓ 1 -minimization programs) are naturally robust against sparse noise. Our framework simplifies and generalizes the existing analyses including compressed sensing, matrix sensing, quadratic sensing, and bilinear sensing. Motivated by recent work on the stochastic gradient method, we also give some experimentally and theoretically preliminary results about the projected stochastic subgradient method. [ABSTRACT FROM AUTHOR]

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ 0 norm. Specifically, the ℓ 0 model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ 0 norm regularization term. Such an ℓ 0 model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the ℓ 0 models found by the proposed inexact algorithm outperform global minimizers of the corresponding ℓ 1 models, in terms of approximation accuracy and sparsity of the solutions. [ABSTRACT FROM AUTHOR]

Application problems can often not be solved adequately by numerical algorithms as several difficulties might arise at the same time. When developing and improving algorithms which hopefully allow to handle those difficulties in the future, good test instances are required. These can then be used to detect the strengths and weaknesses of different algorithmic approaches. In this paper we present a generator for test instances to evaluate solvers for multiobjective mixed-integer linear and nonlinear optimization problems. Based on test instances for purely continuous and purely integer problems with known efficient solutions and known nondominated points, suitable multiobjective mixed-integer test instances can be generated. The special structure allows to construct instances scalable in the number of variables and objective functions. Moreover, it allows to control the resulting efficient and nondominated sets as well as the number of efficient integer assignments. [ABSTRACT FROM AUTHOR]

In multi-objective mixed-integer convex optimization, multiple convex objective functions need to be optimized simultaneously while some of the variables are restricted to take integer values. In this paper, we present a new algorithm to compute an enclosure of the nondominated set of such optimization problems. More precisely, we decompose the multi-objective mixed-integer convex optimization problem into several multi-objective continuous convex optimization problems, which we refer to as patches. We then dynamically compute and improve coverages of the nondominated sets of those patches to finally combine them to obtain an enclosure of the nondominated set of the multi-objective mixed-integer convex optimization problem. Additionally, we introduce a mechanism to reduce the number of patches that need to be considered in total. Our new algorithm is the first of its kind and guaranteed to return an enclosure of prescribed quality within a finite number of iterations. For selected numerical test instances we compare our new criterion space based approach to other algorithms from the literature and show that much larger instances can be solved with our new algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a partial Bregman alternating direction method of multipliers (ADMM) with a general relaxation factor α ∈ (0 , 1 + 5 2) is proposed for structured nonconvex and nonsmooth optimization, where the objective function is the sum of a nonsmooth convex function and a smooth nonconvex function without coupled variables. We add a Bregman distance to alleviate the difficulty of solving the nonsmooth subproblem. For the smooth subproblem, we directly perform a gradient descent step of the augmented Lagrangian function, which makes the computational cost of each iteration of our method very cheap. To our knowledge, the nonconvex ADMM with a relaxation factor α ≠ 1 in the literature has never been studied for the problem under consideration. Under some mild conditions, the boundedness of the generated sequence, the global convergence and the iteration complexity are established. The numerical results verify the efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

We are considering differential-algebraic equations with embedded optimization criteria (DAEOs), in which the embedded optimization problem is solved by global optimization. This leads to differential inclusions for cases in which there are multiple global optimizers at the same time. Jump events from one global optimum to another result in nonsmooth DAEs and reduce the order of convergence of the numerical integrator to first-order. Implementation of event detection and location as introduced in this work preserves the higher-order convergence behaviour of the integrator. This allows the computation of discrete tangent and adjoint sensitivities for optimal control problems. [ABSTRACT FROM AUTHOR]

This paper proposes and establishes the iteration complexity of an inexact proximal accelerated augmented Lagrangian (IPAAL) method for solving linearly constrained smooth nonconvex composite optimization problems. Each IPAAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a suitable Lagrange multiplier update. For any given (possibly infeasible) initial point and tolerance ρ > 0 , it is shown that IPAAL generates an approximate stationary solution in O (ρ - 3 log (ρ - 1)) ACG iterations, which can be improved to O (ρ - 2.5 log (ρ - 1)) if it is further assumed that a certain Slater condition holds. [ABSTRACT FROM AUTHOR]

In this paper, some concepts of convex analysis on hyperbolic spaces are studied. We first study properties of the intrinsic distance, for instance, we present the spectral decomposition of its Hessian. Next, we study the concept of convex sets and the intrinsic projection onto these sets. We also study the concept of convex functions and present first- and second-order characterizations of these functions, as well as some optimization concepts related to them. An extensive study of the hyperbolically convex quadratic functions is also presented. [ABSTRACT FROM AUTHOR]

Semidefinite programming (SDP) problems typically utilize a constraint of the form X ⪰ x x T to obtain a convex relaxation of the condition X = x x T , where x ∈ R n . In this paper, we consider a new hyperplane branching method for SDP based on using an eigenvector of X - x x T . This branching technique is related to previous work of Saxeena et al. (Math Prog Ser B 124:383–411, 2010, https://doi.org/10.1007/s10107-010-0371-9) who used such an eigenvector to derive a disjunctive cut. We obtain excellent computational results applying the new branching technique to difficult instances of the two-trust-region subproblem. [ABSTRACT FROM AUTHOR]

Some properties of σ-convex functions are studied. Then some relations between the notions of σ-subdifferentials and σ-convex functions are established. Also, we present some results regarding the sum formula for the σ-subdifferential. Moreover, we obtain some particular relationships between the σ-subdifferential and the $ (\sigma,y) $ (σ , y) -conjugate. Finally, by imposing some assumptions on f and σ, the maximal $ 2\sigma $ 2 σ -monotonicity of σ-subdifferential is studied. Indeed, the maximality of the usual Fenchel subdifferentials of lower semicontinuous convex functions follows from our result. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate Lagrange saddle points and calmness properties for a nonconvex multiobjective optimization problem by virtue of image regularity conditions. Following along with the image space analysis, the quasi-interior, quasi-relative interior and linear image regularity conditions are presented. Simultaneously, some equivalent characterizations to Fritz John and Karush/Kuhn-Tucker saddle points are established respectively by virtue of the proposed image regularity conditions. Moreover, some calmness properties are also obtained by means of local conic image regularity conditions in the image space of the multiobjective optimization problem. [ABSTRACT FROM AUTHOR]

We introduce a new approach to apply the boosted difference of convex functions algorithm (BDCA) for solving non-convex and non-differentiable problems involving difference of two convex functions (DC functions). Supposing the first DC component differentiable and the second one possibly non-differentiable, the main idea of BDCA is to use the point computed by the subproblem of the DC algorithm (DCA) to define a descent direction of the objective from that point, and then a monotone line search starting from it is performed in order to find a new point which decreases the objective function when compared with the point generated by the subproblem of DCA. This procedure improves the performance of the DCA. However, if the first DC component is non-differentiable, then the direction computed by BDCA can be an ascent direction and a monotone line search cannot be performed. Our approach uses a non-monotone line search in the BDCA (nmBDCA) to enable a possible growth in the objective function values controlled by a parameter. Under suitable assumptions, we show that any cluster point of the sequence generated by the nmBDCA is a critical point of the problem under consideration and provides some iteration-complexity bounds. Furthermore, if the first DC component is differentiable, we present different iteration-complexity bounds and prove the full convergence of the sequence under the Kurdyka–Łojasiewicz property of the objective function. Some numerical experiments show that the nmBDCA outperforms the DCA, such as its monotone version. [ABSTRACT FROM AUTHOR]