Complementarity problems are often used to compute equilibria made up of specifically coordinated solutions of different optimization problems. Specific examples are game-theoretic settings like the bimatrix game or energy market models like for electricity or natural gas. While optimization under uncertainties is rather well-developed, the field of equilibrium models represented by complementarity

ABSTRACT We analyse the existence of Nash equilibria for a class of quadratic multi-leader-follower games using the nonsmooth best response function. To overcome the challenge of nonsmoothness, we pursue a smoothing approach resulting in a reformulation as a smooth Nash equilibrium problem. The existence and uniqueness of solutions are proven for all smoothing parameters. Accumulation points of Nash

Utility-based shortfall risk measure (SR) has received increasing attentions over the past few years. Recently Delage et al. [Shortfall Risk Models When Information of Loss Function Is Incomplete, GERAD HEC, Montréal, 2018] consider a situation where a decision maker's true loss function in the definition of SR is unknown but it is possible to elicit a set of plausible utility functions with partial

ABSTRACT Given the inability to foresee all possible scenarios, it is justified to desire an efficient trust-region subproblem solver capable of delivering any desired level of accuracy on demand; that is, the accuracy obtainable for a given trust-region subproblem should not be partially dependent on the problem itself. Current state-of-the-art iterative eigensolvers all fall into the class of restarted

Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order d ¯ − 1 can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function | ⋅ | apart from arithmetic operations and d ¯ times continuously differentiable

Among many approaches to increase the computational efficiency of semidefinite programming (SDP) relaxation for nonconvex quadratic constrained quadratic programming problems (QCQPs), exploiting the aggregate sparsity of the data matrices in the SDP by Fukuda et al. [Exploiting sparsity in semidefinite programming via matrix completion I: General framework, SIAM J. Optim. 11(3) (2001), pp. 647–674]

In this paper, we propose a new framework to implement interior point method (IPM) in order to solve some very large-scale linear programs (LPs). Traditional IPMs typically use Newton's method to approximately solve a subproblem that aims to minimize a log-barrier penalty function at each iteration. Due its connection to Newton's method, IPM is often classified as second-order method – a genre that

In this paper, we consider the problem of finding ε-approximate stationary points of convex functions that are p-times differentiable with ν-Hölder continuous pth derivatives. We present tensor methods with and without acceleration. Specifically, we show that the non-accelerated schemes take at most O ϵ − 1 / ( p + ν − 1 ) iterations to reduce the norm of the gradient of the objective below given ϵ

In this paper, we consider Newton projection method for solving the quadratic programming problem that emerges in simulation of joining process for assembly with compliant parts. This particular class of problems has specific features such as an ill-conditioned Hessian and a sparse matrix of constraints as well as a requirement for the large-scale computations. We use the projected Newton method with

This paper deals with minimax fractional programs whose objective functions are the maximum of finite ratios of convex functions, with arbitrary convex constraints set. For such problems, Dinkelbach-type algorithms fail to work since the parametric subproblems may be nonconvex, whereas the latter need a global optimal solution of these subproblems. We give necessary optimality conditions for such problems

Discrete convex functions are used in many areas, including operations research, discrete-event systems, game theory, and economics. The objective of this paper is to investigate basic operations such as direct sum, splitting, and aggregation that are related to network induction of discrete convex functions as well as discrete convex sets. Various kinds of discrete convex functions in discrete convex

The precedence constrained traveling salesman (TSP-PC), also known as sequential ordering problem (SOP), consists of finding an optimal tour that satisfies the namesake constraints. Mixed integer-linear programming works well with the ‘lightly constrained’ TSP-PCs, close to asymmetric TSP, as well as the with the ‘heavily constrained’ (Gouveia, Ruthmair, 2015). Dynamic programming (DP) works well with

We develop and analyse a variant of the SARAH algorithm, which does not require computation of the exact gradient. Thus this new method can be applied to general expectation minimization problems rather than only finite sum problems. While the original SARAH algorithm, as well as its predecessor, SVRG, requires an exact gradient computation on each outer iteration, the inexact variant of SARAH (iSARAH)

This paper examines a class of cooperative differential games with continuous updating. Here it is assumed that at each time instant players have or use information about the game structure defined for a closed time interval with fixed duration. The current time continuously evolves with the updating interval. The main problems considered in a cooperative setting with continuous updating is how to

In this paper, we consider a joint-inversion problem using different types of geophysical data: gravity and magnetism. We first formulate two kinds of inverse problems in the famework of the first kind Fredholm integral equations, and then build up a sparse inversion model combining the two inverse problems as well as the cross-gradient term. The cyclic gradient method for quadratic function minimization

This paper studies a class of two-stage distributionally robust optimization (TDRO) problems which comes from many practical application fields. In order to set up some implementable solution method, we first transfer the TDRO problem to its equivalent robust counterpart (RC) by the duality theorem of optimization. The RC reformulation of TDRO is a semi-infinite stochastic programming. Then we construct

We introduce COCO, an open-source platform for Comparing Continuous Optimizers in a black-box setting. COCO aims at automatizing the tedious and repetitive task of benchmarking numerical optimization algorithms to the greatest possible extent. The platform and the underlying methodology allow to benchmark in the same framework deterministic and stochastic solvers for both single and multiobjective

This work presents a partitioned solution procedure to compute shape gradients in fluid–structure interaction (FSI) using black-box adjoint solvers. Special attention is paid to project the gradients onto the undeformed configuration due to the mixed Lagrangian–Eulerian formulation of large-deformation FSI in this work. The adjoint FSI problem is partitioned as an assembly of well-known adjoint fluid

In this paper, we propose a primal-dual interior point trust-region method for solving nonlinear semidefinite programming problems. The method consists of the outer iteration (SDPIP) that finds a Karush–Kuhn–Tucker (KKT) point and the inner iteration (SDPTR) that calculates an approximate barrier KKT point. Algorithm SDPTR combines a commutative class of Newton-like directions with the steepest descent

We introduce the class of multistage stochastic optimization problems with a random number of stages. For such problems, we show how to write dynamic programming equations and how to solve these equations using the Stochastic Dual Dynamic Programming algorithm. Finally, we consider a portfolio selection problem over an optimization period of random duration. For several instances of this problem, we

An extensible open-source deterministic global optimizer (EAGO) programmed entirely in the Julia language is presented. EAGO was developed to serve the need for supporting higher-complexity user-defined functions (e.g. functions defined implicitly via algorithms) within optimization models. EAGO embeds a first-of-its-kind implementation of McCormick arithmetic in an Evaluator structure allowing for

We consider the distributionally robust optimization and show that computing the distributional worst-case is equivalent to computing the projection onto the canonical simplex with additional linear inequality. We consider several distance functions to measure the distance of distributions. We write the projections as optimization problems and show that they are equivalent to finding a zero of real-valued

Mathematical programs with vanishing constraints (MPVCs) are a class of nonlinear optimization problems with applications to various engineering problems such as truss topology design and robot motion planning. MPVCs are difficult problems from both a theoretical and numerical perspective: the combinatorial nature of the vanishing constraints often prevents standard constraint qualifications and optimality

A method for evaluating lexicographical directional (LD)-derivatives of functional programs is presented, extending previous methods to programs containing conditional branches and loops. A language for imperative programs is given, and conditions under which LD-derivatives can be calculated automatically for conditional branches and loops are described, along with a full description of the source

The complexity of solving feasibility problems is considered in this work. It is assumed that the constraints that define the problem can be divided into expensive and cheap constraints. At each iteration, the introduced method minimizes a regularized pth-order model of the sum of squares of the expensive constraints subject to the cheap constraints. Under a Hölder continuity property with constant

Extended formulations of ( k , l ) -sparsity matroids defined on graphs with n vertices and m edges are investigated by Iwata et al. [Extended formulations for sparsity matroids, Math. Program. 158 (2016), pp. 565–574]. This note interprets results on ( k , l ) -lower-truncated transversal polymatroids by the first author in 1983, from the viewpoint of extended formulations, and shows the same O (

For the Lagrangian-DNN relaxation of quadratic optimization problems (QOPs), we propose a Newton-bracketing method to improve the performance of the bisection-projection method implemented in BBCPOP [ACM Tran. Softw., 45(3):34 (2019)]. The relaxation problem is converted into the problem of finding the largest zero y ∗ of a continuously differentiable (except at y ∗ ) convex function g : R → R such

In this paper, we propose a new multilevel Levenberg–Marquardt optimizer for the training of artificial neural networks with quadratic loss function. This setting allows us to get further insight into the potential of multilevel optimization methods. Indeed, when the least squares problem arises from the training of artificial neural networks, the variables subject to optimization are not related by

The paper presents a new approach to solve multifacility location problems, which is based on mixed integer programming and algorithms for minimizing differences of convex (DC) functions. The main challenges for solving the multifacility location problems under consideration come from their intrinsic discrete, nonconvex, and nondifferentiable nature. We provide a reformulation of these problems as

An outerplanar graph is a graph that can be drawn on the plane without crossing edges so that all vertices are on the infinite face. Most organic compounds have outerplanar graph structures. The number of matchings in the skeleton graphs of organic compounds is known as the topological index Z, introduced in the early 70s to investigate correlation between molecular structures and physical properties

Sum-of-squares (SOS) tensors plays an important role in tensor positive definiteness and polynomial optimization. So it is important to figure out what kind of tensors are SOS tensors. In this paper, we first show that several types of even order symmetric tensors are SOS tensors. The inclusive relation between several types of existing SOS tensors are developed under suitable conditions. By exploring

In this work, motivated by the challenging task of learning a deep neural network, we consider optimization problems that consist of minimizing a finite-sum of non-convex and non-smooth functions, where the non-smoothness appears as the maximum of non-convex functions with Lipschitz continuous gradient. Due to the large size of the sum, in practice, we focus here on stochastic first-order methods and

Finding the symmetric rank-1 approximation to a given symmetric tensor is an important problem due to its wide applications and its close relationship to the Z-eigenpair of a tensor. In this paper, we propose a method based on the proximal alternating linearized minimization to directly solve the optimization problem. Global convergence of our algorithm is established. Numerical experiments show that

In this paper, we compute Z-eigenvalues of a class of tensors by studying the properties of semi-symmetric tensor. We prove that Axm−1 is identical to zero if and only if As=0, where As is the associated semi-symmetric tensor of A. Based on the semi-symmetric property, an almost nonnegative irreducible tensor is defined. And we use Newton method to compute Z-eigenvalues of this kind of tensor. The

We consider a nonconvex and nonsmooth group sparse optimization problem where the penalty function is the sum of compositions of a folded concave function and the ℓ2 vector norm for each group variable. We show that under some mild conditions a first-order directional stationary point is a strict local minimizer that fulfils the first-order growth condition, and a second-order directional stationary

This paper studies stochastic optimization problems with polynomials. We propose an optimization model with sample averages and perturbations. The Lasserre-type Moment-SOS relaxations are used to solve the sample average optimization. Properties of the optimization and its relaxations are studied. Numerical experiments are presented.

We derive bounds for the objective errors and gradient residuals when finding approximations to the solution of common regularized quadratic optimization problems within evolving Krylov spaces. These provide upper bounds on the number of iterations required to achieve a given stated accuracy. We illustrate the quality of our bounds on given test examples.

We expand the scope of the alternating direction method of multipliers (ADMM). Specifically, we show that ADMM, when employed to solve problems with multiaffine constraints that satisfy certain verifiable assumptions, converges to the set of constrained stationary points if the penalty parameter in the augmented Lagrangian is sufficiently large. When the Kurdyka–Łojasiewicz (K–Ł) property holds, this

An adaptive regularization algorithm is proposed that uses Taylor models of the objective of order p, p≥2, of the unconstrained objective function, and that is guaranteed to find a first- and second-order critical point in at most O(max{ϵ1−p+1p,ϵ2−p+1p−1}) function and derivatives evaluations, where ϵ1 and ϵ2 are prescribed first- and second-order optimality tolerances. This is a simple algorithm and

It has long been known that the gradient (steepest descent) method may fail on non-smooth problems, but the examples that have appeared in the literature are either devised specifically to defeat a gradient or subgradient method with an exact line search or are unstable with respect to perturbation of the initial point. We give an analysis of the gradient method with steplengths satisfying the Armijo

(2020). Preface. Optimization Methods and Software: Vol. 35, Part I of the special issue dedicated to the 60th birthday of Professor Ya-xiang Yuan. Guest-Editors: Yu-Hong Dai, Xin Liu, Jiawang Nie, Zaiwen Wen, pp. 221-222.

ABSTRACT In this paper, we study the auxiliary problems that appear in p-order tensor methods for unconstrained minimization of convex functions with ν-Hölder continuous pth derivatives. This type of auxiliary problems corresponds to the minimization of a (p+ν)-order regularization of the pth-order Taylor approximation of the objective. For the case p = 3, we consider the use of Gradient Methods with

We introduce exponential augmented Lagrangian methods for solving equilibrium problems in finite-dimensional spaces, extending the so-called quadratic augmented Lagrangian methods. Unlike the quadratic augmented Lagrangian methods that are at most first-order differentiable, our exponential augmented Lagrangian methods can be differentiable at any order. Therefore, second-order methods, such as Newton's

In this paper, we introduce a class of SOR-like iteration methods for solving the systems of the weakly nonlinear equation, which is by reformulating equivalently the weakly nonlinear equation as a two-by-two block nonlinear equation. Two types of the global convergence theorems are given under suitable choices of the involved splitting matrix and parameter. Numerical results for the three-dimensional

We study dual-based algorithms for distributed convex optimization problems over networks, where the objective is to minimize a sum ∑i=1mfi(z) of functions over in a network. We provide complexity bounds for four different cases, namely: each function fi is strongly convex and smooth, each function is either strongly convex or smooth, and when it is convex but neither strongly convex nor smooth. Our

Training generative adversarial networks (GANs) often suffers from cyclic behaviours of iterates. Based on a simple intuition that the direction of centripetal acceleration of an object moving in uniform circular motion is toward the centre of the circle, we present the Simultaneous Centripetal Acceleration (SCA) method and the Alternating Centripetal Acceleration (ACA) method to alleviate the cyclic

This work considers the minimization of a sum of an expectation-valued coordinate-wise smooth nonconvex function and a nonsmooth block-separable convex regularizer. We propose an asynchronous variance-reduced algorithm, where in each iteration, a single block is randomly chosen to update its estimates by a proximal variable sample-size stochastic gradient scheme, while the remaining blocks are kept

Algencan is a well established safeguarded Augmented Lagrangian algorithm introduced in [R. Andreani, E. G. Birgin, J. M. Martínez, and M. L. Schuverdt, On Augmented Lagrangian methods with general lower-level constraints, SIAM J. Optim. 18 (2008), pp. 1286–1309]. Complexity results that report its worst-case behaviour in terms of iterations and evaluations of functions and derivatives that are necessary

A class of direct search methods for locally minimizing a Lipschitz continuous black-box function f subject to locally Lipschitz constraints is presented. A sequence of smoothed ℓ1 penalty functions is used. Each smoothed penalty function is approximately minimized in turn. The smoothing is reduced after each minimization, exposing the ℓ1 exact penalty function in the limit. Convergence to a constrained

Professor Masao Iri and his students had several important contributions to the theory of electric networks. He was one of the first scientists who recognized the importance of matroid theory in studying the qualitative problems of electric networks. The present paper summarizes his results in the unique solvability of linear active networks and in the existence of hybrid immittance description of

The one-to-one correspondence between finite distributive lattices and finite partially ordered sets (posets) is a well-known theorem of G. Birkhoff. This implies a nice representation of any distributive lattice by its corresponding poset, where the size of the former (distributive lattice) is often exponential in the size of the underlying set of the latter (poset). A lot of engineering and economic

We propose and analyse a nonmonotone quasi-Newton algorithm for unconstrained strongly convex multiobjective optimization. In our method, we allow for the decrease of a convex combination of recent function values. We establish the global convergence and local superlinear rate of convergence under reasonable assumptions. We implement our scheme in the context of BFGS quasi-Newton method for solving

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 minimization. While there may exist several affine invariant descent directions for smooth functions at a given point, it is shown that for quadratic functions, there exists exactly one invariant descent direction

The paper illustrates connections between classical results of Analysis and Optimization. The focus is on new elementary proofs of Implicit Function Theorem, Lusternik's Theorem, and optimality conditions for equality constrained optimization problems. The proofs are based on Fermat's Theorem and the Weierstrass Theorem and do not use the contraction mapping principle or other advanced results of Real

We analyze the rate of convergence of the proximal method of multipliers for non-convex nonlinear programming problems. First, we prove, under the strict complementarity condition, that the rate of convergence of the proximal method of multipliers is linear and the ratio constant is proportional to 1/c when the ratio ‖ ( μ 0 , λ 0 ) − ( μ ¯ , λ ¯ ) ‖ / c is small enough, which implies that the rate

We introduce an Alternating Direction Method of Multipliers (ADMM) for finding a solution of the nonsymmetric Eigenvalue Complementarity Problem (EiCP). A simpler version of this method is proposed for the symmetric EiCP, that is, for the computation of a Stationary Point (SP) of a Standard Fractional Quadratic Program. The algorithm is also extended for the computation of an SP of a Standard Quadratic

In this paper, we propose a projection-free accelerated method for solving convex optimization problems with unbounded feasible set. The method is an accelerated gradient scheme such that each projection subproblem is approximately solved by means of a conditional gradient scheme. Under reasonable assumptions, it is shown that an ϵ-approximate solution (concept related to the optimal value of the problem)

We present a feasible full step interior-point algorithm to solve the P∗(κ) horizontal linear complementarity problem defined on a Cartesian product of symmetric cones, which is not based on a usual barrier function. The full steps are scaled utilizing the Nesterov-Todd (NT) scaling point. Our approach generates the search directions leading to the full-NT steps by algebraically transforming the centring

We consider two-machine routing open shop problem on a tree. In this problem, a transportation network with a tree-like structure is given, and each node contains some jobs to be processed by two mobile machines. Machines are initially located in the predefined node called the depot, have to traverse the network to perform their operations on each job (and each job has to be processed by both machines

This paper proposes two algorithms that are based on a subgradient and an inertial scheme with the explicit iterative method for solving pseudomonotone equilibrium problems. The weak convergence of both algorithms is well-established under standard assumptions on the cost bifunction. The advantage of these algorithms is that they did not require any line search procedure or any knowledge about bifunction