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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]

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]

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]

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]

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]

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]

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]

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]