The problem of recovering acoustic sources, more specifically monopoles, from point-wise measurements of the corresponding acoustic pressure at a limited number of frequencies is addressed. To this purpose, a family of sparse optimization problems in measure space in combination with the Helmholtz equation on a bounded domain is considered. A weighted norm with unbounded weight near the observation

The exterior Bernoulli problem is rephrased into a shape optimization problem using a new type of objective function called the Dirichlet-data-gap cost function which measures the \(L^2\)-distance between the Dirichlet data of two state functions. The first-order shape derivative of the cost function is explicitly determined via the chain rule approach. Using the same technique, the second-order shape

The alternating direction method of multipliers (ADMM) is widely used to solve separable convex programming problems. At each iteration, the classical ADMM solves the subproblems exactly. For many problems arising from practical applications, it is usually impossible or too expensive to obtain the exact solution of a subproblem. To overcome this, a special proximal term is added to ease the solvability

In this work we study binary two-stage robust optimization problems with objective uncertainty. We present an algorithm to calculate efficiently lower bounds for the binary two-stage robust problem by solving alternately the underlying deterministic problem and an adversarial problem. For the deterministic problem any oracle can be used which returns an optimal solution for every possible scenario

We consider a misspecified optimization problem, requiring the minimization of a function \(f(\cdot;\theta ^*)\) over a closed and convex set X where \(\theta ^*\) is an unknown vector of parameters that may be learnt by a parallel learning process. Here, we develop coupled schemes that generate iterates \((x_k,\theta _k)\) as \(k \rightarrow \infty\), then \(x_k \rightarrow x^*\), a minimizer of \(f(\cdot;\theta

In this paper, we propose an adaptation of the classical augmented Lagrangian method for dealing with multi-objective optimization problems. Specifically, after a brief review of the literature, we give a suitable definition of Augmented Lagrangian for equality and inequality constrained multi-objective problems. We exploit this object in a general computational scheme that is proved to converge, under

In a series of papers (Solodov and Svaiter in J Convex Anal 6(1):59–70, 1999; Set-Valued Anal 7(4):323–345, 1999; Numer Funct Anal Optim 22(7–8):1013–1035, 2001) Solodov and Svaiter introduced new inexact variants of the proximal point method with relative error tolerances. Point-wise and ergodic iteration-complexity bounds for one of these methods, namely the hybrid proximal extragradient method (1999)

We propose a regularization method for nonlinear least-squares problems with equality constraints. Our approach is modeled after those of Arreckx and Orban (SIAM J Optim 28(2):1613–1639, 2018. https://doi.org/10.1137/16M1088570) and Dehghani et al. (INFOR Inf Syst Oper Res, 2019. https://doi.org/10.1080/03155986.2018.1559428) and applies a selective regularization scheme that may be viewed as a reformulation

Convex and nonconvex finite-sum minimization arises in many scientific computing and machine learning applications. Recently, first-order and second-order methods where objective functions, gradients and Hessians are approximated by randomly sampling components of the sum have received great attention. We propose a new trust-region method which employs suitable approximations of the objective function

This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2):518–538, 1993) and using occupation measures, the optimal control problem is written into a linear programming problem of infinite-dimension (weak formulation). Thanks to Interval arithmetic

In this paper, we describe a new active-set algorithmic framework for minimizing a non-convex function over the unit simplex. At each iteration, the method makes use of a rule for identifying active variables (i.e., variables that are zero at a stationary point) and specific directions (that we name active-set gradient related directions) satisfying a new “nonorthogonality” type of condition. We prove

Quasi-convex optimization acts a pivotal part in many fields including economics and finance; the subgradient method is an effective iterative algorithm for solving large-scale quasi-convex optimization problems. In this paper, we investigate the quantitative convergence theory, including the iteration complexity and convergence rates, of various subgradient methods for solving quasi-convex optimization

This paper proposes and analyzes an inexact variant of the proximal generalized alternating direction method of multipliers (ADMM) for solving separable linearly constrained convex optimization problems. In this variant, the first subproblem is approximately solved using a relative error condition whereas the second one is assumed to be easy to solve. In many ADMM applications, one of the subproblems

This short note considers an efficient variant of the trust-region algorithm with dynamic accuracy proposed by Carter (SIAM J Sci Stat Comput 14(2):368–388, 1993) and by Conn et al. (Trust-region methods. MPS-SIAM series on optimization, SIAM, Philadelphia, 2000) as a tool for very high-performance computing, an area where it is critical to allow multi-precision computations for keeping the energy

In this paper we consider the classical unconstrained nonlinear multiobjective optimization problem. For such a problem, it is particularly interesting to compute as many points as possible in an effort to approximate the so-called Pareto front. Consequently, to solve the problem we define an “a posteriori” algorithm whose generic iterate is represented by a set of points rather than by a single one

The characterization and numerical solution of two non-smooth optimal control problems governed by a Fokker–Planck (FP) equation are investigated in the framework of the Pontryagin maximum principle (PMP). The two FP control problems are related to the problem of determining open- and closed-loop controls for a stochastic process whose probability density function is modelled by the FP equation. In

Euclidean distance matrix optimization with ordinal constraints (EDMOC) has found important applications in sensor network localization and molecular conformation. It can also be viewed as a matrix formulation of multidimensional scaling, which is to embed n points in a r-dimensional space such that the resulting distances follow the ordinal constraints. The ordinal constraints, though proved to be

This paper proposes an efficient adaptive variant of a quadratic penalty accelerated inexact proximal point (QP-AIPP) method proposed earlier by the authors. Both the QP-AIPP method and its variant solve linearly set constrained nonconvex composite optimization problems using a quadratic penalty approach where the generated penalized subproblems are solved by a variant of the underlying AIPP method

The problem of minimizing the sum of a convex quadratic function and the ratio of two quadratic functions can be reformulated as a Celis–Dennis–Tapia (CDT) problem and, thus, according to some recent results, can be polynomially solved. However, the degree of the known polynomial approaches for these problems is fairly large and that justifies the search for efficient local search procedures. In this

We propose a discretization algorithm for solving a class of nonsmooth convex semi-infinite programming problems that is based on a bundle method. Instead of employing the inexact calculation to evaluate the lower level problem, we shall carry out a discretization scheme. The discretization method is used to get a number of discretized problems which are solved by the bundle method. In particular,

We extend the result on the spectral projected gradient method by Birgin et al. in 2000 to a log-determinant semidefinite problem with linear constraints and propose a spectral projected gradient method for the dual problem. Our method is based on alternate projections on the intersection of two convex sets, which first projects onto the box constraints and then onto a set defined by a linear matrix

Mathematical programs with or-constraints form a new class of disjunctive optimization problems with inherent practical relevance. In this paper, we provide a comparison of three different solution methods for the numerical treatment of this problem class which are inspired by classical approaches from disjunctive programming. First, we study the replacement of the or-constraints as nonlinear inequality

In this article a special class of nonlinear optimal control problems involving a bilinear term in the boundary condition is studied. These kind of problems arise for instance in the identification of an unknown space-dependent Robin coefficient from a given measurement of the state, or when the Robin coefficient can be controlled in order to reach a desired state. Necessary and sufficient optimality

We introduce new global and local inexact oracle concepts for a wide class of convex functions in composite convex minimization. Such inexact oracles naturally arise in many situations, including primal–dual frameworks, barrier smoothing, and inexact evaluations of gradients and Hessians. We also provide examples showing that the class of convex functions equipped with the newly inexact oracles is

This paper considers the efficient minimization of the infinite time average of a stationary ergodic process in the space of a handful of design parameters which affect it. Problems of this class, derived from physical or numerical experiments which are sometimes expensive to perform, are ubiquitous in engineering applications. In such problems, any given function evaluation, determined with finite

This paper is concerned with the stochastic structured tensors to stochastic complementarity problems. The definitions and properties of stochastic structured tensors, such as the stochastic strong P-tensors, stochastic P-tensors, stochastic \(P_{0}\)-tensors, stochastic strictly semi-positive tensors and stochastic S-tensors are given. It is shown that the expected residual minimization formulation

A strongly orthogonal decomposition of a tensor is a rank one tensor decomposition with the two component vectors in each mode of any two rank one tensors are either colinear or orthogonal. A strongly orthogonal decomposition with few number of rank one tensors is favorable in applications, which can be represented by a matrix-tensor multiplication with orthogonal factor matrices and a sparse tensor;

In this paper, we study the higher-degree tensor eigenvalue complementarity problem (HDTEiCP). We give an upper bound for the number of the higher-degree complementarity eigenvalues for the generic HDTEiCP. A semidefinite relaxation algorithm is proposed for computing all the higher-degree complementarity eigenvalues sequentially, as well as the corresponding eigenvectors, and the convergence of the

The purpose of this paper is to study the problem of computing unitary eigenvalues (U-eigenvalues) of non-symmetric complex tensors. By means of symmetric embedding of complex tensors, the relationship between U-eigenpairs of a non-symmetric complex tensor and unitary symmetric eigenpairs (US-eigenpairs) of its symmetric embedding tensor is established. An Algorithm 3.1 is given to compute the U-eigenvalues

\(\mathbf {P}\)-tensor and \(\mathbf {P}_0\)-tensor are introduced in tensor complementarity problem, which have wide applications in game theory. In this paper, we establish SDP relaxation algorithms for detecting \(\mathbf {P}(\mathbf {P}_0)\)-tensor. We first reformulate \(\mathbf {P}(\mathbf {P}_0)\)-tensor detection problem as polynomial optimization problems. Then we propose the SDP relaxation

Tensor decompositions have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-one outer products using either the CANDECOMP/PARAFAC, the Tucker model, or some variations thereof. The motivation of these decompositions is to find an approximate representation for a given tensor. The main propose of this paper is to develop two neural

This paper discusses how to compute all real solutions of the second-order cone tensor complementarity problem when there are finitely many ones. For this goal, we first formulate the second-order cone tensor complementarity problem as two polynomial optimization problems. Based on the reformulation, a semidefinite relaxation method is proposed by solving a finite number of semidefinite relaxations

This paper concerns saddle points of rational functions, under general constraints. Based on optimality conditions, we propose an algorithm for computing saddle points. It uses Lasserre’s hierarchy of semidefinite relaxation. The algorithm can get a saddle point if it exists, or it can detect its nonexistence if it does not. Numerical experiments show that the algorithm is efficient for computing saddle

This paper presents a generalization of the spectral norm and the nuclear norm of a tensor via arbitrary tensor partitions, a much richer concept than block tensors. We show that the spectral p-norm and the nuclear p-norm of a tensor can be lower and upper bounded by manipulating the spectral p-norms and the nuclear p-norms of subtensors in an arbitrary partition of the tensor for \(1\le p\le \infty\)

We consider the semi-infinite system of polynomial inequalities of the form $$\begin{aligned} {{\mathbf {K}}}:=\{x\in {{\mathbb {R}}}^m\mid p(x,y)\ge 0,\quad \forall y\in S\subseteq {{\mathbb {R}}}^n\}, \end{aligned}$$ where p(x, y) is a real polynomial in the variables x and the parameters y, the index set S is a basic semialgebraic set in \({{\mathbb {R}}}^n\), \(-p(x,y)\) is convex in x for every

We investigate the convergence of a recently popular class of first-order primal–dual algorithms for saddle point problems under the presence of errors in the proximal maps and gradients. We study several types of errors and show that, provided a sufficient decay of these errors, the same convergence rates as for the error-free algorithm can be established. More precisely, we prove the (optimal) \(O\left(

This paper considers a class of two-stage stochastic linear variational inequality problems whose first stage problems are stochastic linear box-constrained variational inequality problems and second stage problems are stochastic linear complementary problems having a unique solution. We first give conditions for the existence of solutions to both the original problem and its perturbed problems. Next

Abstract The multi-objective spanning tree (MoST) is an extension of the minimum spanning tree problem (MST) that, as well as its single-objective counterpart, arises in several practical applications. However, unlike the MST, for which there are polynomial-time algorithms that solve it, the MoST is NP-hard. Several researchers proposed techniques to solve the MoST, each of those methods with specific

Abstract We study a facility location problem where a single facility serves multiple customers each represented by a (possibly non-convex) region in the plane. The aim of the problem is to locate a single facility in the plane so that the maximum of the closest Euclidean distances between the facility and the customer regions is minimized. Assuming that each customer region is mixed-integer second

Abstract Many real-world applications can usually be modeled as convex quadratic problems. In the present paper, we want to tackle a specific class of quadratic programs having a dense Hessian matrix and a structured feasible set. We hence carefully analyze a simplicial decomposition like algorithmic framework that handles those problems in an effective way. We introduce a new master solver, called

Abstract In this paper, we introduce and analyze a new algorithm for solving equilibrium problem involving pseudomonotone and Lipschitz-type bifunction in real Hilbert space. The algorithm requires only a strongly convex programming problem per iteration. A weak and a strong convergence theorem are established without the knowledge of the Lipschitz-type constants of the bifunction. As a special case

Abstract The alternating direction method of multipliers (ADMM) is being widely used in a variety of areas; its different variants tailored for different application scenarios have also been deeply researched in the literature. Among them, the linearized ADMM has received particularly wide attention in many areas because of its efficiency and easy implementation. To theoretically guarantee convergence

Abstract Quasi-Newton methods are often used in the frame of non-linear optimization. In those methods, the quality and cost of the estimate of the Hessian matrix has a major influence on the efficiency of the optimization algorithm, which has a huge impact for computationally costly problems. One strategy to create a more accurate estimate of the Hessian consists in maximizing the use of available

Abstract We consider a cutting-plane algorithm for solving mixed-integer semidefinite optimization (MISDO) problems. In this algorithm, the positive semidefinite (psd) constraint is relaxed, and the resultant mixed-integer linear optimization problem is solved repeatedly, imposing at each iteration a valid inequality for the psd constraint. We prove the convergence properties of the algorithm. Moreover

Abstract In this paper, we present a line search exact penalty method for solving nonlinear semidefinite programming (SDP) problem. Compared with the traditional sequential semidefinite programming (SSDP) method which requires that the subproblem at every iterate point is compatible, this method is more practical. We first use a robust subproblem, which is always feasible, to get a detective step,

Nonlinear disjunctive convex sets arise naturally in the formulation or solution methods of many discrete–continuous optimization problems. Often, a tight algebraic representation of the disjunctive convex set is sought, with the tightest such representation involving the characterization of the convex hull of the disjunctive convex set. In the most general case, this can be explicitly expressed through

This paper studies an acceleration technique for incremental aggregated gradient (IAG) method through the use of curvature information for solving strongly convex finite sum optimization problems. These optimization problems of interest arise in large-scale learning applications. Our technique utilizes a curvature-aided gradient tracking step to produce accurate gradient estimates incrementally using

In this paper, we propose an Augmented Lagrangian algorithm for solving a general class of possible non-convex problems called quasi-equilibrium problems (QEPs). We define an Augmented Lagrangian bifunction associated with QEPs, introduce a secondary QEP as a measure of infeasibility and we discuss several special classes of QEPs within our theoretical framework. For obtaining global convergence under

Our contribution in this paper is twofold. First, the global convergence analysis of the recently proposed pattern search implicit filtering algorithm (PSIFA), aimed at linearly constrained noisy minimization problems, is revisited to address more general locally Lipschitz objective functions corrupted by noise. Second, PSIFA is applied for solving the damped harmonic oscillator parameter identification

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

This paper considers the problem of minimizing an expectation function over a closed convex set, coupled with a function or expectation constraint on either decision variables or problem parameters. We first present a new stochastic approximation (SA) type algorithm, namely the cooperative SA (CSA), to handle problems with the constraint on devision variables. We show that this algorithm exhibits the

We consider a class of optimistic bilevel problems. Specifically, we address bilevel problems in which at the lower level the objective function is fully convex and the feasible set does not depend on the upper level variables. We show that this nontrivial class of mathematical programs is sufficiently broad to encompass significant real-world applications and proves to be numerically tractable. From

We study the convergence of a modified proximal point method for DC functions in Hadamard manifolds. We use the iteration computed by the proximal point method for DC function extended to the Riemannian context by Souza and Oliveira (J Glob Optim 63:797–810, 2015) to define a descent direction which improves the convergence of the method. Our method also accelerates the classical proximal point method

Interior point methods have attracted most of the attention in the recent decades for solving large scale convex quadratic programming problems. In this paper we take a different route as we present an augmented Lagrangian method for convex quadratic programming based on recent developments for nonlinear programming. In our approach, box constraints are penalized while equality constraints are kept

We develop new variants of Benders decomposition methods for variational inequality problems. The construction is done by applying the general class of Dantzig–Wolfe decomposition of Luna et al. (Math Program 143(1–2):177–209, 2014) to an appropriately defined dual of the given variational inequality, and then passing back to the primal space. As compared to previous decomposition techniques of the

The elementary Euclidean concept of circumcenter has recently been employed to improve two aspects of the classical Douglas–Rachford method for projecting onto the intersection of affine subspaces. The so-called circumcentered–reflection method is able to both accelerate the average reflection scheme by the Douglas–Rachford method and cope with the intersection of more than two affine subspaces. We

We propose an SQP algorithm for mathematical programs with vanishing constraints which solves at each iteration a quadratic program with linear vanishing constraints. The algorithm is based on the newly developed concept of [Formula: see text]-stationarity (Benko and Gfrerer in Optimization 66(1):61-92, 2017). We demonstrate how [Formula: see text]-stationary solutions of the quadratic program can

In this article, we introduce the Generalized [Formula: see text]-Survivable Network Design Problem ([Formula: see text]-GSNDP) which has applications in the design of backbone networks. Different mixed integer linear programming formulations are derived by combining previous results obtained for the related [Formula: see text]-GSNDP and Generalized Network Design Problems. An extensive computational

Classical penalty methods solve a sequence of unconstrained problems that put greater and greater stress on meeting the constraints. In the limit as the penalty constant tends to ∞, one recovers the constrained solution. In the exact penalty method, squared penalties are replaced by absolute value penalties, and the solution is recovered for a finite value of the penalty constant. In practice, the

First principles approaches to the protein structure prediction problem must search through an enormous conformational space to identify low-energy, near-native structures. In this paper, we describe the formulation of the tertiary structure prediction problem as a nonlinear constrained minimization problem, where the goal is to minimize the energy of a protein conformation subject to constraints on