Search

This paper proposes a path-based approach for the minimization of a continuously differentiable function over sparse symmetric sets, which is a hard problem that exhibits a restrictiveness-hierarchy of necessary optimality conditions. To achieve the more restrictive conditions in the hierarchy, state-of-the-art algorithms require a support optimization oracle that must exactly solve the problem in smaller dimensions. The path-based approach developed in this study produces a path-based optimality condition, which is placed well in the restrictiveness-hierarchy, and a method to achieve it that does not require a support optimization oracle and, moreover, is projection-free. In the development process, new results are derived for the regularized linear minimization problem over sparse symmetric sets, which give additional means to identify optimal solutions for convex and concave objective functions. We complement our results with numerical examples. [ABSTRACT FROM AUTHOR]

Recent papers indicate that some algorithms for constrained optimization may exhibit worst-case complexity bounds that are very similar to those of unconstrained optimization algorithms. A natural question is whether well-established practical algorithms, perhaps with small variations, may enjoy analogous complexity results. In the present paper we show that the answer is positive with respect to inexact restoration algorithms in which first-order approximations are employed for defining the subproblems. [ABSTRACT FROM AUTHOR]

This paper aims at developing two versions of the generalized Newton method to compute local minimizers for nonsmooth problems of unconstrained and constrained optimization that satisfy an important stability property known as tilt stability. We start with unconstrained minimization of continuously differentiable cost functions having Lipschitzian gradients and suggest two second-order algorithms of Newton type: one involving coderivatives of Lipschitzian gradient mappings, and the other based on graphical derivatives of the latter. Then we proceed with the propagation of these algorithms to minimization of extended-real-valued prox-regular functions, while covering in this way problems of constrained optimization, by using Moreau envelopes. Employing advanced techniques of second-order variational analysis and characterizations of tilt stability allows us to establish the solvability of subproblems in both algorithms and to prove the Q-superlinear convergence of their iterations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a primal-dual algorithm with a novel momentum term using the partial gradients of the coupling function that can be viewed as a generalization of the method proposed by Chambolle and Pock in 2016 to solve saddle point problems defined by a convex-concave function L(x,y)=f(x)+Φ(x,y)--h(y) with a general coupling term Φ(x,y) that is not assumed to be bilinear. Assuming ∇xΦ(*, y) is Lipschitz for any fixed y, and ∇yΦ(*, *) is Lipschitz, we show that the iterate sequence converges to a saddle point; and for any (x,y), we derive error bounds in terms of ... for the ergodic sequence ... In particular, we show O(1/k) rate when the problem is merely convex in x. Furthermore, assuming Φ(x,*) is linear for each fixed x and f is strongly convex, we obtain the ergodic convergence rate of O(1/k²) -- we are not aware of another single-loop method in the related literature achieving the same rate when Φ is not bilinear. Finally, we propose a backtracking technique which does not require the knowledge of Lipschitz constants while ensuring the same convergence results. We also consider convex optimization problems with nonlinear functional constraints and we show that using the backtracking scheme, the optimal convergence rate can be achieved even when the dual domain is unbounded. We tested our method against other state-of-the-art first-order algorithms and interior-point methods for solving quadratically constrained quadratic problems with synthetic data, the kernel matrix learning, and regression with fairness constraints arising in machine learning. [ABSTRACT FROM AUTHOR]

In this paper, we propose two novel non-stationary first-order primal-dual algorithms to solve non-smooth composite convex optimization problems. Unlike existing primal-dual schemes where the parameters are often fixed, our methods use predefined and dynamic sequences for parameters. We prove that our first algorithm can achieve an O(1/k) convergence rate on the primal-dual gap, and primal and dual objective residuals, where k is the iteration counter. Our rate is on the non-ergodic (i.e., the last iterate) sequence of the primal problem and on the ergodic (i.e., the averaging) sequence of the dual problem, which we call the semi-ergodic rate. By modifying the step-size update rule, this rate can be boosted even faster on the primal objective residual. When the problem is strongly convex, we develop a second primal-dual algorithm that exhibits an O(1/k²) convergence rate on the same three types of guarantees. Again by modifying the step-size update rule, this rate becomes faster on the primal objective residual. Our primal-dual algorithms are the first ones to achieve such fast convergence rate guarantees under mild assumptions compared to existing works, to the best of our knowledge. As byproducts, we apply our algorithms to solve constrained convex optimization problems and prove the same convergence rates on both the objective residuals and the feasibility violation. We still obtain at least O(1/k²) rates even when the problem is "semi-strongly" convex. We verify our theoretical results via two well-known numerical examples. [ABSTRACT FROM AUTHOR]

In many inverse problems, a nonnegativity constraint is natural. Moreover, in some cases, we expect the vector of unknown parameters to have zero components. When a Bayesian approach is taken, this motivates a desire for prior probability density (and hence posterior probability density) functions that have positive mass at the boundary of the set {x∊RN|x&#88040}. Unfortunately, it is difficult to define a prior with this property that yields computationally tractable inference for large-scale inverse problems. In this paper, we use nonnegativity constrained optimization to define such prior and posterior density functions when the measurement error is either Gaussian or Poisson distributed. The numerical optimization methods we use are highly efficient, and hence our approach is computationally tractable even in large-scale cases. We embed our nonnegativity constrained optimization approach within a hierarchical framework, obtaining Gibbs samplers for both Gaussian and Poisson distributed measurement cases. Finally, we test the resulting Markov chain Monte Carlo methods on examples from both image deblurring and positron emission tomography. [ABSTRACT FROM AUTHOR]

This paper considers strategies for selecting the barrier parameter at every iteration of an interior-point method for nonlinear programming. Numerical experiments suggest that heuristic adaptive choices, such as Mehrotra's probing procedure, outperform monotone strategies that hold the barrier parameter fixed until a barrier optimality test is satisfied. A new adaptive strategy is proposed based on the minimization of a quality function. The paper also proposes a globalization framework that ensures the convergence of adaptive interior methods, and examines convergence failures of the Mehrotra predictor-corrector algorithm. The barrier update strategies proposed in this paper are applicable to a wide class of interior methods and are tested in the two distinct algorithmic frameworks provided by the IPOPT and KNITRO software packages. [ABSTRACT FROM AUTHOR]

In the first part of this two-paper series, we presented a conceptual algorithm for the optimal control of constrained switched systems and proved that this algorithm generates a sequence of points that converge to a necessary condition for optimality. However, since our algorithm requires the exact solution of a differential equation, the numerical implementation of this algorithm is impractical. In this paper, we address this shortcoming by constructing an implementable algorithm that discretizes the differential equation, producing a finite-dimensional nonlinear program. We prove that this implementable algorithm constructs a sequence of points that asymptotically satisfy a necessary condition for optimality for the constrained switched system optimal control problem. Four simulation experiments are included to validate the theoretical developments. [ABSTRACT FROM AUTHOR]

Classical primal-dual algorithms attempt to solve max∣1 miι⅛ £(æ, μ) by alternately minimizing over the primal variable x through primal descent and maximizing the dual variable μ through dual ascent. However, when Z2(x, μ) is highly nonconvex with complex constraints in x, the minimization over x may not achieve global optimality and, hence, the dual ascent step loses its valid intuition. This observation motivates us to propose a new class of primal-dual algorithms for nonconvex constrained optimization with the key feature to reverse dual ascent to a conceptually new dual descent, in a sense, elevating the dual variable to the same status as the primal variable. Surprisingly, this new dual scheme achieves some best iteration complexities for solving nonconvex optimization problems. In particular, when the dual descent step is scaled by a fractional constant, we name it scaled dual descent (SDD), otherwise, unsealed dual descent (UDD). For nonconvex multiblock optimization with nonlinear equality constraints, we propose SDD-alternating direction method of multipliers (SDD-ADMM) and show that it finds an e-stationary solution in O{e~4) iterations. The complexity is further improved to O{e~3) and O{e~2) under proper conditions. We also propose UDD-augmented Lagrangian method (UDD-ALM), combining UDD with ALM, for weakly convex minimization over affine constraints. We show that UDD-ALM finds an e-stationary solution in O{e~2) iterations. These complexity bounds for both algorithms either achieve or improve the best-known results in the ADMM and ALM literature. Moreover, SDD-ADMM addresses a longstanding limitation of existing ADMM frameworks. [ABSTRACT FROM AUTHOR]

In this paper we consider inequality constrained nonlinear optimization problems where the first order derivatives of the objective function and the constraints cannot be used. Our starting point is the possibility to transform the original constrained problem into an unconstrained or linearly constrained minimization of a nonsmooth exact penalty function. This approach shows two main difficulties: the first one is the nonsmoothness of this class of exact penalty functions which may cause derivative-free codes to converge to nonstationary points of the problem; the second one is the fact that the equivalence between stationary points of the constrained problem and those of the exact penalty function can only be stated when the penalty parameter is smaller than a threshold value which is not known a priori. In this paper we propose a derivative-free algorithm which overcomes the preceding difficulties and produces a sequence of points that admits a subsequence converging to a Karush--Kuhn--Tucker point of the constrained problem. In particular the proposed algorithm is based on a smoothing of the nondifferentiable exact penalty function and includes an updating rule which, after at most a finite number of updates, is able to determine a "right value" for the penalty parameter. Furthermore we present the results obtained on a real world problem concerning the estimation of parameters in an insulin-glucose model of the human body. [ABSTRACT FROM AUTHOR]

In this paper we discuss the polynomiality of a feasible version of Mehrotra's predictor-corrector algorithm whose variants have been widely used in several interior point method (IPM)-based optimization packages. A numerical example is given that shows that the adaptive choice of centering parameter and correction terms in this algorithm may lead to small steps being taken in order to keep the iterates in a large neighborhood of the central path, which is important for proving polynomial complexity properties of this method. Motivated by this example, we introduce a safeguard in Mehrotra's algorithm that keeps the iterates in the prescribed neighborhood and allows us to obtain a positive lower bound on the step size. This safeguard strategy is also used when the affine scaling direction performs poorly. We prove that the safeguarded algorithm will terminate after at most O(n2log(x°}T s° /e) iterations. By modestly modifying the corrector direction, we reduce the iteration complexity to O(nlog(x°}T s° /e). To ensure fast asymptotic convergence of the algorithm, we changed Mehrotra's updating scheme of the centering parameter slightly while keeping the safeguard. The new algorithms have the same order of iteration complexity as the safeguarded algorithms but enjoy superlinear convergence as well. Numerical results using the McIPM and LIPSOL software packages are reported. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of minimization of a nonsmooth function under general nonsmooth constraints when no derivatives of the objective or constraint functions are available. We introduce the mesh adaptive direct search (MADS) class of algorithms which extends the generalized pattern search (GPS) class by allowing local exploration, called polling, in an asymptotically dense set of directions in the space of optimization variables. This means that under certain hypotheses, including a weak constraint qualification due to Rockafellar, MADS can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained. The main GPS convergence result is to identify limit points x0302;, where the Clarke generalized derivatives are nonnegative in a finite set of directions, called refining directions. Although in the unconstrained case, nonnegative combinations of these directions span the whole space, the fact that there can only be finitely many GPS refining directions limits rigorous justification of the barrier approach to finitely many linear constraints for GPS. The main result of this paper is that the general MADS framework is flexible enough to allow the generation of an asymptotically dense set of refining directions along which the Clarke derivatives are nonnegative. We propose an instance of MADS for which the refining directions are dense in the hypertangent cone at x0302; with probability 1 whenever the iterates associated with the refining directions converge to a single x0302;. The instance of MADS is compared to versions of GPS on some test problems. We also illustrate the limitation of our results with examples. [ABSTRACT FROM AUTHOR]

An algorithm is proposed to solve robust control problems constrained by partial differential equations with uncertain coefficients, based on the so-called MG/OPT framework. The levels in the MG/OPT hierarchy correspond to discretization levels of the PDE, as usual. For stochastic problems, the relevant quantities (such as the gradient) contain expected value operators on each of these levels. They are estimated using a multilevel Monte Carlo method, the specifics of which depend on the MG/OPT level. Each of the optimization levels then contains multiple underlying multilevel Monte Carlo levels. The MG/OPT hierarchy allows the algorithm to exploit the structure inherent in the PDE, speeding up the convergence to the optimum. In contrast, the multilevel Monte Carlo hierarchy exists to exploit structure present in the stochastic dimensions of the problem. A statement about the asymptotic cost of the algorithm is proven, and some additional properties are discussed. The performance of the algorithm is numerically investigated for three test cases. A reduction in the number of samples required on expensive levels and therefore in computation time can be observed. [ABSTRACT FROM AUTHOR]

The alternating direction method of multipliers (ADMM) is widely used to solve large-scale linearly constrained optimization problems, convex or nonconvex, in many engineering fields. However there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex. In this paper we analyze the convergence of the ADMM for solving certain nonconvex consensus and sharing problems. We show that the classical ADMM converges to the set of stationary solutions, provided that the penalty parameter in the augmented Lagrangian is chosen to be sufficiently large. For the sharing problems, we show that the ADMM is convergent regardless of the number of variable blocks. Our analysis does not impose any assumptions on the iterates generated by the algorithm and is broadly applicable to many ADMM variants involving proximal update rules and various flexible block selection rules. [ABSTRACT FROM AUTHOR]

In [SIAM J. Optim., 17 (2006), pp. 188-217] Audet and Dennis proposed the class of mesh adaptive direct search (MADS) algorithms for minimization of a nonsmooth function under general nonsmooth constraints. The notation used in the paper evolved since the preliminary versions, and, unfortunately, even though the statement of Proposition 4.2 is correct, it is not compatible with the final notation. The purpose of this note is to show that the proposition is valid. [ABSTRACT FROM AUTHOR]

The Douglas{Rachford algorithm (DRA) is a powerful optimization method for minimizing the sum of two convex (not necessarily smooth) functions. The vast majority of previous research dealt with the case when the sum has at least one minimizer. In the absence of minimizers, it was recently shown that for the case of two indicator functions, the DRA converges to a best approximation solution. In this paper, we present a new convergence result on the DRA applied to the problem of minimizing a convex function subject to a linear constraint. Indeed, a normal solution may be found even when the domain of the objective function and the linear subspace constraint have no point in common. As an important application, a new parallel splitting result is provided. We also illustrate our results through various examples. [ABSTRACT FROM AUTHOR]

Various effective algorithms have been proposed in the past two decades for nonlinear PDEs arising from the unconstrained total-variation-based image denoising problem regularizing the total variation constrained minimization model. Such algorithms can be used to obtain a satisfactory result as long as a suitable regularization parameter balancing the trade-off between a good fit to the data and a regular solution is given. However, it is generally difficult to obtain a suitable regularization parameter without which restored images can be unsatisfactory: if it is too large, then the resulting solution is still contaminated by noise, while if too small, the solution is a poor approximation of the true noise-free solution. To provide an automatic method for the regularization parameter when the noise level is known a priori, one way is to address the coupled Karush--Kuhn-- Tucker (KKT) systems from the constrained total variation optimization problem. So far much less work has been done on this problem. This paper presents an iterative update algorithm for a Lagrange multiplier to solve the KKT conditions, and our proposed method can adaptively deal with noisy images with different variances δ². Numerical experiments show that our model can effectively find a highly accurate solution and produce excellent restoration results in terms of image quality. [ABSTRACT FROM AUTHOR]

In this paper, we propose an interior-point method for linearly constrained|and possibly nonconvex|optimization problems. The method|which we call the Hessian barrier algorithm (HBA)|combines a forward Euler discretization of Hessian-Riemannian gradient flows with an Armijo backtracking step-size policy. In this way, HBA can be seen as an alternative to mirror descent, and contains as special cases the affine scaling algorithm, regularized Newton processes, and several other iterative solution methods. Our main result is that, modulo a nondegeneracy condition, the algorithm converges to the problem's critical set; hence, in the convex case, the algorithm converges globally to the problem's minimum set. In the case of linearly constrained quadratic programs (not necessarily convex), we also show that the method's convergence rate is O(1/k p) for some p ε (0,1] that depends only on the choice of kernel function (i.e., not on the problem's primitives). These theoretical results are validated by numerical experiments in standard nonconvex test functions and large-scale traffic assignment problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new way of choosing directions for the mesh adaptive direct search (MADS) class of algorithms. The advantages of this new ORTHOMADS instantiation of MADS are that the polling directions are chosen deterministically, ensuring that the results of a given run are repeatable, and that they are orthogonal to each other, which yields convex cones of missed directions at each iteration that are minimal in a reasonable measure. Convergence results for ORTHOMADS follow directly from those already published for MADS, and they hold deterministically, rather than with probability one, as is the case for LTMADS, the first MADS instance. The initial numerical results are quite good for both smooth and nonsmooth and constrained and unconstrained problems considered here. [ABSTRACT FROM AUTHOR]

This paper describes a new algorithm for solving nonlinear programming problems with equality constraints. The method introduces the idea of using trust cylinders to keep the infeasibility under control. Each time the trust cylinder is violated, a restoration step is called and the infeasibility level is reduced. The radius of the trust cylinder has a nonincreasing update scheme, so eventually a feasible (and optimal) point is obtained. Global and local convergence of the algorithm are analyzed, as well as its numerical performance. The results suggest that the algorithm is promising. [ABSTRACT FROM AUTHOR]

We describe an asynchronous parallel derivative-free algorithm for linearly constrained optimization. Generating set search (GSS) is the basis of our method. At each iteration, a GSS algorithm computes a set of search directions and corresponding trial points and then evaluates the objective function value at each trial point. Asynchronous versions of the algorithm have been developed in the unconstrained and bound-constrained cases which allow the iterations to continue (and new trial points to be generated and evaluated) as soon as any other trial point completes. This enables better utilization of parallel resources and a reduction in overall run time, especially for problems where the objective function takes minutes or hours to compute. For linearly constrained GSS, the convergence theory requires that the set of search directions conforms to the nearby boundary. This creates an immediate obstacle for asynchronous methods where the definition of nearby is not well defined. In this paper, we develop an asynchronous linearly constrained GSS method that overcomes this difficulty and maintains the original convergence theory. We describe our implementation in detail, including how to avoid function evaluations by caching function values and using approximate lookups. We test our implementation on every CUTEr test problem with general linear constraints and up to 1000 variables. Without tuning to individual problems, our implementation was able to solve 95% of the test problems with 10 or fewer variables, 73% of the problems with 11-100 variables, and nearly half of the problems with 100-1000 variables. To the best of our knowledge, these are the best results that have ever been achieved with a derivative-free method for linearly constrained optimization. Our asynchronous parallel implementation is freely available as part of the APPSPACK software. [ABSTRACT FROM AUTHOR]

Global convergence in constrained optimization algorithms has traditionally been enforced by the use of parametrized penalty functions. Recently, the filter strategy has been introduced as an alternative. At least part of the motivation for using filter methods consists of avoiding the need for estimating a suitable penalty parameter, which is often a delicate task. In this paper, we demonstrate that the use of a parametrized penalty function in nonsmooth convex optimization can be avoided without using the relatively complex filter methods. We propose an approach which appears to be more direct and easier to implement, in the sense that it is closer in spirit and structure to the well-developed unconstrained bundle methods. Preliminary computational results are also reported. [ABSTRACT FROM AUTHOR]

This paper formulates and analyzes a pattern search method for general constrained optimization based on filter methods for step acceptance. Roughly, a filter method accepts a step that improves either the objective function value or the value of some function that measures the constraint violation. The new algorithm does not compute or approximate any derivatives, penalty constants, or Lagrange multipliers. A key feature of the new algorithm is that it preserves the division into search and local poll steps, which allows the explicit use of inexpensive surrogates or random search heuristics in the search step. It is shown here that the algorithm identifies limit points at which optimality conditions depend on local smoothness of the functions and, to a greater extent, on the choice of a certain set of directions. Stronger optimality conditions are guaranteed for smoother functions and, in the constrained case, for a fortunate choice of the directions on which the algorithm depends. These directional conditions generalize those given previously for linear constraints, but they do not require a feasible starting point. In the absence of general constraints, the proposed algorithm and its convergence analysis generalize previous work on unconstrained, bound constrained, and linearly constrained generalized pattern search. The algorithm is illustrated on some test examples and on an industrial wing platform engineering design application. [ABSTRACT FROM AUTHOR]

An optimization algorithm for nonsmooth nonconvex constrained optimization problems with upper-C² objective functions is proposed and analyzed. Upper-C² is a weakly concave property that exists in difference of convex (DC) functions and arises naturally in many applications, particularly certain classes of solutions to parametric optimization problems e.g., recourse of stochastic programming and projection onto closed sets. The algorithm can be viewed as an extension of sequential quadratic programming (SQP) to nonsmooth problems with upper-² objectives or a simplified bundle method. It is globally convergent with bounded algorithm parameters that are updated with a trust-region criterion. The algorithm handles general smooth constraints through linearization and uses a line search to ensure progress. The potential inconsistencies from the linearization of the constraints are addressed through a penalty method. The capabilities of the algorithm are demonstrated by solving both simple upper-² problems and a real-world optimal power flow problem used in current power grid industry practices. [ABSTRACT FROM AUTHOR]

This paper describes a new trust region method for solving large-scale optimization problems with nonlinear equality and inequality constraints. The new algorithm employs interiorpoint techniques from linear programming, adapting them for more general nonlinear problems. A software implementation based entirely on sparse matrix methods is described. The software handles infeasible start points, identifies the active set of constraints at a solution, and can use second derivative information to solve problems. Numerical results are reported for large and small problems, and a comparison is made with other large-scale optimization codes. [ABSTRACT FROM AUTHOR]

A method is proposed for solving equality constrained nonlinear optimization problems involving twice continuously differentiable functions. The method employs a trust funnel approach consisting of two phases: a first phase to locate an ε-feasible point and a second phase to seek optimality while maintaining at least ε-feasibility. Two-phase approaches of this kind based on a cubic regularization methodology have recently been proposed along with supporting worst-case iteration complexity analyses. Notably, in these approaches, the objective function is completely ignored in the first phase when ε-feasibility is sought. The main contribution of the method proposed in this paper is that the same worst-case iteration complexity is achieved, but with a first phase that also accounts for improvements in the objective function. As such, the method attempts to put significantly less burden on the second phase for seeking optimality. [ABSTRACT FROM AUTHOR]

We present a framework for analyzing convergence and local rates of convergence of a class of descent algorithms, assuming the objective function is weakly convex. The framework is general, in the sense that it combines the possibility of explicit iterations (based on the gradient or a subgradient at the current iterate), implicit iterations (using a subgradient at the next iteration, like in the proximal schemes), as well as iterations when the associated subgradient is specially constructed and does not correspond either to the current or the next point (this is the case of descent steps in bundle methods). Under the subdifferential-based error bound on the distance to critical points, linear rates of convergence are established. Our analysis applies, among other techniques, to prox-descent for decomposable functions, the proximal-gradient method for a sum of functions, redistributed bundle methods, and a class of algorithms that can be cast in the feasible descent framework for constrained optimization. [ABSTRACT FROM AUTHOR]

We propose and analyze a pseudotransient continuation algorithm for dynamics on subsets of RN. Examples include certain flows on manifolds and the dynamic formulation of bound-constrained optimization problems. The method gets its global convergence properties from the dynamics and inherits its local convergence properties from any fast locally convergent iteration. [ABSTRACT FROM AUTHOR]

We present an algorithm for large-scale equality constrained optimization. The method is based on a characterization of inexact sequential quadratic programming (SQP) steps that can ensure global convergence. Inexact SQP methods are needed for large-scale applications for which the iteration matrix cannot be explicitly formed or factored and the arising linear systems must be solved using iterative linear algebra techniques. We address how to determine when a given inexact step makes sufficient progress toward a solution of the nonlinear program, as measured by an exact penalty function. The method is globalized by a line search. An analysis of the global convergence properties of the algorithm and numerical results are presented. [ABSTRACT FROM AUTHOR]

Sequential quadratic optimization algorithms are proposed for solving smooth nonlinear optimization problems with equality constraints. The main focus is an algorithm proposed for the case when the constraint functions are deterministic, and constraint function and derivative values can be computed explicitly, but the objective function is stochastic. It is assumed in this setting that it is intractable to compute objective function and derivative values explicitly, although one can compute stochastic function and gradient estimates. As a starting point for this stochastic setting, an algorithm is proposed for the deterministic setting that is modeled after a state-of-theart line-search SQP algorithm but uses a stepsize selection scheme based on Lipschitz constants (or adaptively estimated Lipschitz constants) in place of the line search. This sets the stage for the proposed algorithm for the stochastic setting, for which it is assumed that line searches would be intractable. Under reasonable assumptions, convergence (resp., convergence in expectation) from remote starting points is proved for the proposed deterministic (resp., stochastic) algorithm. The results of numerical experiments demonstrate the practical performance of our proposed techniques. [ABSTRACT FROM AUTHOR]

The time parallel solution of optimality systems arising in PDE constrained optimization could be achieved by simply applying any time parallel algorithm, such as Parareal, to solve the forward and backward evolution problems arising in the optimization loop. We propose here a different strategy by devising directly a new time parallel algorithm, which we call ParaOpt, for the coupled forward and backward nonlinear partial differential equations. ParaOpt is inspired by the Parareal algorithm for evolution equations and thus is automatically a two-level method. We provide a detailed convergence analysis for the case of linear parabolic PDE constraints. We illustrate the performance of ParaOpt with numerical experiments for both linear and nonlinear optimality systems. [ABSTRACT FROM AUTHOR]

Motivated by the TRACE algorithm [F. E. Curtis, D. P. Robinson, and M. Samadi, Math. Program., 162 (2017), pp. 1--32], we propose a trust region algorithm for finding second-order stationary points of a linearly constrained nonconvex optimization problem. We show the convergence of the proposed algorithm to (∈g, ∈H)-second-order stationary points in O (max{ ∈3-2 g}) iterations. This iteration complexity is achieved for general linearly constrained optimization without cubic regularization of the objective function. [ABSTRACT FROM AUTHOR]-3 H}) iterations. This iteration complexity is achieved for general linearly constrained optimization without cubic regularization of the objective function. [ABSTRACT FROM AUTHOR]

We study constrained nested stochastic optimization problems in which the objective function is a composition of two smooth functions whose exact values and derivatives are not available. We propose a single timescale stochastic approximation algorithm, which we call the nested averaged stochastic approximation (NASA), to find an approximate stationary point of the problem. The algorithm has two auxiliary averaged sequences (filters) which estimate the gradient of the composite objective function and the inner function value. By using a special Lyapunov function, we show that the NASA achieves the sample complexity of O (1/ε² ) for finding an ε -approximate stationary point, thus outperforming all extant methods for nested stochastic approximation. Our method and its analysis are the same for both unconstrained and constrained problems, without any need of batch samples for constrained nonconvex stochastic optimization. We also present a simplified parameterfree variant of the NASA method for solving constrained single-level stochastic optimization problems, and we prove the same complexity result for both unconstrained and constrained problems. [ABSTRACT FROM AUTHOR]

Adaptive cubic regularization methods have emerged as a credible alternative to linesearch and trust-region for smooth nonconvex optimization, with optimal complexity amongst second-order methods. Here we consider a general/new class of adaptive regularization methods that use first- or higher-order local Taylor models of the objective regularized by a(ny) power of the step size and applied to convexly constrained optimization problems. We investigate the worst-case evaluation complexity/global rate of convergence of these algorithms, when the level of sufficient smoothness of the objective may be unknown or may even be absent. We find that the methods accurately reflect in their complexity the degree of smoothness of the objective and satisfy increasingly better bounds with improving model accuracy. The bounds vary continuously and robustly with respect to the regularization power and accuracy of the model and the degree of smoothness of the objective. [ABSTRACT FROM AUTHOR]

We propose a method for constrained and unconstrained nonlinear multiobjective optimization problems that is based on an SQP-type approach. The proposed algorithm maintains a list of nondominated points that is improved both for spread along the Pareto front and optimality by solving single-objective constrained optimization problems. These single-objective problems are derived as SQP problems based on the given nondominated points. Under appropriate differentiability assumptions we discuss convergence to local optimal Pareto points. We provide numerical results for a set of unconstrained and constrained multiobjective optimization problems in the form of performance and data profiles, where several performance metrics are used. The numerical results confirm the superiority of the proposed algorithm against a state-of-the-art multiobjective solver and a classical scalarization approach, both in the quality of the approximated Pareto front and in the computational effort necessary to compute the approximation. [ABSTRACT FROM AUTHOR]

The evaluation complexity of general nonlinear, possibly nonconvex, constrained optimization is analyzed. It is shown that, under suitable smoothness conditions, an ε-approximate first-order critical point of the problem can be computed in order O(ε1-2(p+1)/p) evaluations of the problem's functions and their first p derivatives. This is achieved by using a two-phase algorithm inspired by Cartis, Gould, and Toint [SIAM J. Optim., 21 (2011), pp. 1721-1739; SIAM J. Optim., 23 (2013), pp. 1553-1574]. It is also shown that strong guarantees (in terms of handling degeneracies) on the possible limit points of the sequence of iterates generated by this algorithm can be obtained at the cost of increased complexity. At variance with previous results, the -approximate first-order criticality is defined by satisfying a version of the KKT conditions with an accuracy that does not depend on the size of the Lagrange multipliers. [ABSTRACT FROM AUTHOR]

We consider a bilevel problem involving two monotone equilibrium bifunctions and we show that this problem can be solved by a proximal point method with generalized proximal distances. We propose a framework for the convergence analysis of the sequence generated by the algorithm. This class of problems is very interesting because it covers mathematical programs and optimization problems under equilibrium constraints. As an application, we consider the problem of the stability and change dynamics of a leader-follower relationship in a hierarchical organization. [ABSTRACT FROM AUTHOR]

Joint chance constrained problems give rise to many algorithmic challenges. Even in the convex case, i.e., when an appropriate transformation of the probabilistic constraint is a convex function, its cutting-plane linearization is just an approximation, produced by an oracle providing subgradient and function values that can only be evaluated inexactly. As a result, the cutting-plane model may lie above the true constraint. For dealing with such upper inexact oracles, and still solving the problem up to certain precision, a special numerical algorithm must be put in place. We introduce a family of constrained bundle methods, based on the so-called improvement functions, that is shown to be convergent and encompasses many previous approaches as well as new algorithms. Depending on the oracle accuracy, we analyze to what extent the considered methods solve the joint chance constrained program. The approach is assessed on real-life energy problems, arising when dealing with stochastic hydroreservoir management. [ABSTRACT FROM AUTHOR]

We consider optimization problems with objective and constraint functions that may be nonconvex and nonsmooth. Problems of this type arise in important applications, many having solutions at points of nondifferentiability of the problem functions. We present a line search algorithm for situations when the objective and constraint functions are locally Lipschitz and continuously differentiable on open dense subsets of Rn. Our method is based on a sequential quadratic programming (SQP) algorithm that uses an l1 penalty to regularize the constraints. A process of gradient sampling (GS) is employed to make the search direction computation effective in nonsmooth regions. We prove that our SQP-GS method is globally convergent to stationary points with probability one and illustrate its performance with a MATLAB implementation. [ABSTRACT FROM AUTHOR]

Numerous scientific applications across a variety of fields depend on box-constrained convex optimization. Box-constrained problems therefore continue to attract research interest. We address box-constrained (strictly convex) problems by deriving two new quasi-Newton algorithms. Our algorithms are positioned between the projected-gradient [J. B. Rosen, J. SIAM, 8 (1960), pp. 181-217] and projected-Newton [D. P. Bertsekas, SIAM J. Control Optim., 20 (1982), pp. 221- 246] methods. We also prove their convergence under a simple Armijo step-size rule. We provide experimental results for two particular box-constrained problems: nonnegative least squares (NNLS), and nonnegative Kullback-Leibler (NNKL) minimization. For both NNLS and NNKL our algorithms perform competitively as compared to well-established methods on medium-sized problems; for larger problems our approach frequently outperforms the competition. [ABSTRACT FROM AUTHOR]

We present a line-search algorithm for large-scale continuous optimization. The algorithm is matrix-free in that it does not require the factorization of derivative matrices. Instead, it uses iterative linear system solvers. Inexact step computations are supported ill order to save computational expense during each iteration. The algorithm is an interior-point approach derived from an inexact Newton method for equality constrained optimization proposed by Curtis, Nocedal, and Wächter [SIAM J. Optim., 20 (2009), pp. 1224 1249], with additional functionality for handling inequality constraints. The algorithm is shown to be globally convergent under loose assumptions. Numerical results are presented for nonlinear optimization test set collections and a pair of PDE- constrained model problems. [ABSTRACT FROM AUTHOR]

We develop and compare multilevel algorithms for solving constrained nonlinear variational problems via interior point methods. Several equivalent formulations of the linear systems arising at each iteration of the interior point method are compared from the point of view of conditioning and iterative solution. Furthermore, we show how a multilevel continuation strategy can be used to obtain good initial guesses ("hot starts") for each nonlinear iteration. Some minimal surface and volume-constrained image registration problems are used to illustrate the various approaches. [ABSTRACT FROM AUTHOR]

We present a new “lifting” approach for the solution of nonlinear optimization problems (NLPs) that have objective and constraint functions with intermediate variables. Introducing these as additional degrees of freedom into the original problem, combined with adding suitable new constraints to ensure equivalence of the problems, we propose to solve this augmented system instead of the original system by a Newton-type method. This often offers advantages in terms of convergence rates and region of attraction. The main contribution of this article is an efficient algorithmic trick to generate the quantities needed for a Newton-type method on the augmented (“lifted”) system with (a) almost no additional computational cost per iteration compared to a nonlifted Newton method, and (b) with negligible programming burden. We derive lifted schemes for Newton’s method, as well as for constrained Gauss–Newton and adjoint based sequential quadratic programming (SQP) methods, and show equivalence of the new efficiently lifted approaches with “full-space” lifted Newton iterations. We establish conditions on the intermediate functions that imply faster local quadratic convergence for lifted versus nonlifted Newton iterations, a phenomenon often observed in practice but not yet explained theoretically. Finally, we compare numerically the behavior of the lifted approach with the nonlifted approach on several test problems, including a large scale example with 27 million intermediate variables. The algorithms and examples are available as open-source code in the C++ package LiftOpt. [ABSTRACT FROM AUTHOR]

We consider active set identification for linearly constrained optimization problems in the absence of explicit information about the derivative of the objective function. We begin by presenting some general results on active set identification that are not tied to any particular algorithm. These general results are sufficiently strong that, given a sequence of iterates converging to a Karush–Kuhn–Tucker point, it is possible to identify binding constraints for which there are nonzero multipliers. We then focus on generating set search methods, a class of derivative-free direct search methods. We discuss why these general results, which are posed in terms of the direction of steepest descent, apply to generating set search, even though these methods do not have explicit recourse to derivatives. Nevertheless, there is a clearly identifiable subsequence of iterations at which we can reliably estimate the set of constraints that are binding at a solution. We discuss how active set estimation can be used to accelerate generating set search methods and illustrate the appreciable improvement that can result using several examples from the CUTEr test suite. We also introduce two algorithmic refinements for generating set search methods. The first expands the subsequence of iterations at which we can make inferences about stationarity. The second is a more flexible step acceptance criterion. [ABSTRACT FROM AUTHOR]

An optimal control problem governed by an elliptic variational inequality is studied. The feasible set of the problem is relaxed, and a path-following-type method is used to regularize the constraint on the state variable. First order optimality conditions for the relaxed-regularized subproblems are derived, and convergence of stationary points with respect to the relaxation and regularization parameters is shown. In particular, C- and strong stationarity as well as variants thereof are studied. The subproblems are solved by using semismooth Newton methods. The overall algorithmic concept is provided, and its performance is discussed by means of examples, including problems with bilateral constraints and a nonsymmetric operator. [ABSTRACT FROM AUTHOR]

A new fast iterative algorithm is proposed to denoise images affected by Gaussian additive noise, by solving a constrained optimization problem. A family of functional parameterized by a few hyperparameters already proposed in the literature to solve denoising problems is considered. The algorithm estimates jointly the image and the hyperparameters, therefore providing an automatic method. Each member of the family is made up of a coherence with the data term and a term enforcing a roughness penalty and preserving jump discontinuities. Assuming we know the noise variance, an adequacy constraint is also considered. The algorithm computes a member of the family and a minimizer of it which satisfies the constraint. A convergence proof is provided. We then consider a heuristic version of the algorithm which gives restorations of comparable quality whose computational complexity is a linear function of the pixels number. Experimental results on synthetic and real data are presented. Moreover, numerical comparisons with several fast denoising methods are provided. [ABSTRACT FROM AUTHOR]

We consider a large-scale convex minimization problem with nonnegativity constraints that arises in astronomical imaging. We develop a cost functional which incorporates the statistics of the noise in the image data and Tikhonov regularization to induce stability. We introduce an efficient hybrid gradient projection-reduced Newton (active set) method. By “reduced Newton” we mean taking Newton steps only in the inactive variables. Due to the large size of our problem, we compute approximate reduced Newton steps using conjugate gradient (CG) iteration. We also introduce a highly effective sparse preconditioner that dramatically speeds up CG convergence. A numerical comparison between our method and other standard large-scale constrained minimization algorithms is presented. [ABSTRACT FROM AUTHOR]