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The separation of two sets (or more specific of two cones) plays an important role in different fields of mathematics such as variational analysis, convex analysis, convex geometry, and optimization. In the paper, we derive some new results for the separation of two not necessarily convex cones by a (convex) cone/conical surface in real (topological) linear spaces. Basically, we follow the separation approach by Kasimbeyli [SIAM J. Optim., 20 (2010), pp. 1591-1619] based on augmented dual cones and Bishop--Phelps type (normlinear) separating functions. Classical separation theorems for convex sets are the key tool for proving our main nonlinear cone separation theorems. [ABSTRACT FROM AUTHOR]

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This paper studies an optimization problem on the sum of traces of matrix quadratic forms in m semiorthogonal matrices, which can be considered as a generalization of the synchronization of rotations. While the problem is nonconvex, this paper shows that its semidefinite pro-gramming relaxation solves the original nonconvex problems exactly with high probability under an additive noise model with small noise in the order of O(m1/4). In addition, it shows that, with high probability, the sufficient condition for global optimality considered in Won, Zhou, and Lange [SIAM J. Matrix Anal. Appl., 2 (2021), pp. 859-882] is also necessary under a similar small noise condition. These results can be considered as a generalization of existing results on phase synchronization. [ABSTRACT FROM AUTHOR]

Packing and covering semidefinite programs (SDPs) appear in natural relaxations of many combinatorial optimization problems as well as a number of other applications. Recently, several techniques were proposed, which utilize the particular structure of this class of problems, to obtain more efficient algorithms than those offered by general SDP solvers. For certain applications, such as those described in this paper, it may be desirable to obtain sparse dual solutions, i.e., those with support size (almost) independent of the number of primal constraints. In this paper, we give an algorithm that finds such solutions, which is an extension of a logarithmic-potential based algorithm of Grigoriadis et al. [SIAM J. Optim. 11 (2001), pp. 1081-1091] for packing/covering linear programs. [ABSTRACT FROM AUTHOR]

Given a closed convex set A in a Banach space X, motivated by the continuity and Fréchet differentiability of A introduced, respectively, in [D. Gale and V. Klee, Math. Scand., 7 (1959), pp. 379-391] and [X. Y. Zheng, SIAM J. Optim., 30 (2020), pp. 490-512], this paper considers the Cp-differentiability, subdifferentiability, and G\^ateaux differentiability of A. Using the technique of variational analysis, it is proved that A is Cp-differentiable (resp., subdifferentiable or G\^ateaux differentiable) if and only if for every continuous convex function f : X → R with infx∈A f(x) > infx∈X f(x) the corresponding constrained optimization problem PA(f) is 2/p-order-well-posed solvable (resp., generalized well-posed solvable or weak well-posed solvable). It is also proved that if the conjugate function f* of a continuous convex function f on X is Cp-differentiable on dom(f*), then for every closed convex set A in X with infx∈A f(x) > -∞ the corresponding optimization problem PA(f) is 2/p-order-well-posed solvable. As a byproduct, every constrained convex optimization problem with a strongly convex quadratic objective function is proved to be globally second-order-well-posed solvable. Our main results are new even in the case of finite dimensional spaces. [ABSTRACT FROM AUTHOR]

In this paper, we adopt the augmented Lagrangian method (ALM) to solve convex quadratic second-order cone programming problems (SOCPs). Fruitful results on the efficiency of the ALM have been established in the literature. Recently, it has been shown in [Cui, Sun, and Toh, Math. Program., 178 (2019), pp. 381-415] that if the quadratic growth condition holds at an optimal solution for the dual problem, then the KKT residual converges to zero R-superlinearly when the ALM is applied to the primal problem. Moreover, Cui, Ding, and Zhao [SIAM J. 0ptim., 27 (2017), pp. 2332-2355] provided sufficient conditions for the quadratic growth condition to hold under the metric subregularity and bounded linear regularity conditions for solving composite matrix optimization problems involving spectral functions. Here, we adopt these recent ideas to analyze the convergence properties of the ALM when applied to SOCPs. To the best of our knowledge, no similar work has been done for SOCPs so far. In our paper, we first provide sufficient conditions to ensure the quadratic growth condition for SOCPs. With these elegant theoretical guarantees, we then design an SOCP solver and apply it to solve various classes of SOCPs, such as minimal enclosing ball problems, classical trust-region subproblems, square-root Lasso problems, and DIMACS Challenge problems. Numerical results show that the proposed ALM based solver is efficient and robust compared to the existing highly developed solvers, such as Mosek and SDPT3. [ABSTRACT FROM AUTHOR]

Recently, in a series of papers [Y. Rahimi, C. Wang, H. Dong, and Y. Lou, SIAM J. Sci. Comput., 41 (2019), pp. A3649-A3672; C. Wang, M. Tao, J. Nagy, and Y. Lou, SIAM J. Imaging Sci., 14 (2021), pp. 749-777; C. Wang, M. Yan, and Y. Lou, IEEE Trans. Signal Process., 68 (2020), pp. 2660-2669; P. Yin, E. Esser, and J. Xin, Commun. Inf. Syst., 14 (2014), pp. 87-109], the ratio of ℓ1 and ℓ2 norms was proposed as a sparsity inducing function for noiseless compressed sensing. In this paper, we further study properties of such model in the noiseless setting, and propose an algorithm for minimizing ℓ1/ℓ2 subject to noise in the measurements. Specifically, we show that the extended objective function (the sum of the objective and the indicator function of the constraint set) of the model in [Y. Rahimi, C. Wang, H. Dong, and Y. Lou, SIAM J. Sci. Comput., 41 (2019), pp. A3649-A3672] satisfies the Kurdyka--Lojasiewicz (KL) property with exponent 1/2; this allows us to establish linear convergence of the algorithm proposed in [C. Wang, M. Yan, and Y. Lou, IEEE Trans. Signal Process., 68 (2020), pp. 2660-2669] (see equation 11) under mild assumptions. We next extend the ℓ1/ℓ2 model to handle compressed sensing problems with noise. We establish the solution existence for some of these models under the spherical section property [S. A. Vavasis, Derivation of Compressive Sensing Theorems from the Spherical Section Property, University of Waterloo, 2009; Y. Zhang, J. Oper. Res. Soc. China, 1 (2013), pp. 79-105] and extend the algorithm in [C. Wang, M. Yan, and Y. Lou, IEEE Trans. Signal Process., 68 (2020), pp. 2660-2669] (see equation 11) by incorporating moving-balls-approximation techniques [A. Auslender, R. Shefi, and M. Teboulle, SIAM J. Optim., 20 (2010), pp. 3232-3259] for solving these problems. We prove the subsequential convergence of our algorithm under mild conditions and establish global convergence of the whole sequence generated by our algorithm by imposing additional KL and differentiability assumptions on a specially constructed potential function. Finally, we perform numerical experiments on robust compressed sensing and basis pursuit denoising with residual error measured by \ell 2 norm or Lorentzian norm via solving the corresponding ℓ1/ℓ2 models by our algorithm. Our numerical simulations show that our algorithm is able to recover the original sparse vectors with reasonable accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we study smooth stochastic multilevel composition optimization problems, where the objective function is a nested composition of T functions. We assume access to noisy evaluations of the functions and their gradients, through a stochastic first-order oracle. For solving this class of problems, we propose two algorithms using moving-average stochastic estimates, and analyze their convergence to an e-stationary point of the problem. We show that the first algorithm, which is a generalization of [S. Ghadimi, A. Ruszczynski, and M. Wang, SIAM J. Optim., 30 (2020), pp. 960-979] to the T level case, can achieve a sample complexity of OT(1/ε6) by using minibatches of samples in each iteration, where OT hides constants that depend on T. By modifying this algorithm using linearized stochastic estimates of the function values, we improve the sample complexity to OT(1/ε4). This modification not only removes the requirement of having a minibatch of samples in each iteration, but also makes the algorithm parameter-free and easy to implement. To the best of our knowledge, this is the first time that such an online algorithm designed for the (un)constrained multilevel setting obtains the same sample complexity of the smooth single-level setting, under standard assumptions (unbiasedness and boundedness of the second moments) on the stochastic first-order oracle. [ABSTRACT FROM AUTHOR]

In this paper we provide a generalization of the Douglas--Rachford splitting (DRS) and the primal-dual algorithm [L. Condat, J. Optim. Theory Appl., 158 (2013), pp. 460-479; B. C. V\~u, Adv. Comput. Math., 38 (2013), pp. 667-681] for solving monotone inclusions in a real Hilbert space involving a general linear operator. The proposed method allows for primal and dual nonstandard metrics and activates the linear operator separately from the monotone operators appearing in the inclusion. In the simplest case when the linear operator has full range, it reduces to classical DRS. Moreover, the weak convergence of primal-dual sequences to a Kuhn--Tucker point is guaranteed, generalizing the main result in [B. F. Svaiter, SIAM J. Control Optim., 49 (2011), pp. 280-287]. Inspired by [D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, M. Fortin and R. Glowinski, eds., Stud. Math. Appl. 15, North-Holland, Amsterdam, 1983, pp. 299-331], we also derive a new split alternating direction method of multipliers (SADMM) by applying our method to the dual of a convex optimization problem involving a linear operator which can be expressed as the composition of two linear operators. The proposed SADMM activates one linear operator implicitly and the other one explicitly, and we recover ADMM when the latter is set as the identity. Connections and comparisons of our theoretical results with respect to the literature are provided for the main algorithm and SADMM. The flexibility and efficiency of both methods is illustrated via numerical simulations in total variation image restoration and a sparse minimization problem. [ABSTRACT FROM AUTHOR]

In [V. Guigues, SIAM J. Optim., 30 (2020), pp. 407-438], an inexact variant of stochastic dual dynamic programming (SDDP) called ISDDP was introduced which uses approximate (instead of exact with SDDP) primal dual solutions of the problems solved in the forward and backward passes of the method. That variant of SDDP was studied in [V. Guigues, SIAM J. Optim., 30 (2020), pp. 407-438] for linear and for differentiable nonlinear multistage stochastic programs (MSPs). In this paper, we extend ISDDP to nondifferentiable MSPs. We first provide formulas for inexact cuts for value functions of convex nondifferentiable optimization problems. We then combine these cuts with SDDP to describe ISDDP for nondifferentiable MSPs and analyze the convergence of the method. More precisely, for a problem with T stages, we show that for errors bounded from ab ove by\varepsilon, the limit superior and limit inferior of sequences of upper and lower bounds on the optimal value of the problem are at most at distance 3\varepsilonT to the optimal value and that for asymptotically vanishing errors ISDDP converges to an optimal policy. Finally, we present the results of encouraging numerical experiments on a multistage nondifferentiable stochastic convex program solved using exact SDDP and the proposed ISDDP. [ABSTRACT FROM AUTHOR]

In this paper, we study the sequential convex programming method with monotone line search (SCPls) in [Z. Lu, Sequential Convex Programming Methods for a Class of Structured Nonlinear Programming, https://arxiv.org/abs/1210.3039, 2012] for a class of difference-of-convex optimization problems with multiple smooth inequality constraints. The SCPls is a representative variant of moving-ball-approximation-type algorithms [A. Auslender, R. Shefi, and M. Teboulle, SIAM J. Optim., 20 (2010), pp. 3232-3259; J. Bolte, Z. Chen, and E. Pauwels, Math. Program., 182 (2020) pp. 1-36; J. Bolte and E. Pauwels, Math. Oper. Res., 41 (2016), pp. 442-465; R. Shefi and M. Teboulle, Math. Program., 159 (2016), pp. 137-164] for constrained optimization problems. We analyze the convergence rate of the sequence generated by SCPls in both nonconvex and convex settings by imposing suitable Kurdyka--Löjasiewicz (KL) assumptions. Specifically, in the nonconvex settings, we assume that a special potential function related to the ob jective and the constraints is a KL function, while in the convex settings we impose KL assumptions directly on the extended ob jective function (i.e., sum of the ob jective and the indicator function of the constraint set). A relationship between these two different KL assumptions is established in the convex settings under additional differentiability assumptions. We also discuss how to deduce the KL exponent of the extended ob jective function from its Lagrangian in the convex settings, under additional assumptions on the constraint functions. Thanks to this result, the extended ob jectives of some constrained optimization models such as minimizing\ell1 sub ject to logistic/Poisson loss are found to be KL functions with exponent 1 under mild assumptions. To illustrate how our results can be applied, we consider SCPs for minimizing 1-2 [P-Yin, Y. Lou, Q. He, and J. Xin, SIAM J. Sci. Comput., 37 (2015), pp. A536--A563] subject to residual error measured by norm/Lorentzian norm [R. E. Carrillo, A. B. Ramirez, G. R. Arce, K. E. Barner, and B. M. Sadler, EURASIP J. Adv. Signal Process. (2016), 108]. We first discuss how the various conditions required in our analysis can be verified, and then perform numerical experiments to illustrate the convergence behaviors of SCPls. [ABSTRACT FROM AUTHOR]

In this paper, through an intuitive vanil la proximal method perspective, we derive a concise unified acceleration framework (UAF) for minimizing a convex function that has H\"older continuous derivatives with respect to general (non-Euclidean) norms. The UAF reconciles two different high-order acceleration approaches, one by Nesterov [Math. Program., 112 (2008), pp. 159-181] and one by Monteiro and Svaiter [SIAM J. Optim., 23 (2013), pp. 1092-1125]. As a result, the UAF unifies the high-order acceleration instances of the two approaches by only two problem-related parameters and two additional parameters for framework design. Meanwhile, the UAF (and its analysis) is the first approach to make high-order methods applicable for high-order smoothness conditions with respect to non-Euclidean norms. Furthermore, the UAF is the first approach that can match the existing lower bound of iteration complexity for minimizing a convex function with H\"older continuous derivatives. For practical implementation, we introduce a new and effective heuristic that significantly simplifies the binary search procedure required by the framework. We use experiments on logistic regression to verify the effectiveness of the heuristic. Finally, the UAF is proposed directly in the general composite convex setting and shows that the existing high-order algorithms can be naturally extended to the general composite convex setting. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fundamental problem in modern digital communications known as multiple-input multiple-output (MIMO) detection, which can be formulated as a complex quadratic programming problem subject to unit-modulus and discrete argument constraints. Various semidefinite-relaxation-based (SDR-based) algorithms have been proposed to solve the problem in the literature. In this paper, we first show that conventional SDR is generally not tight for the problem. Then, we propose a new and enhanced SDR and show its tightness under an easily checkable condition, which essentially requires the level of the noise to be below a certain threshold. The above results have answered an open question posed by So in [Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'10), SIAM, Philadelphia, PA, 2011, pp. 698-711]. Numerical simulation results show that our proposed SDR significantly outperforms the conventional SDR in terms of the relaxation gap. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fundamental problem in modern digital communications known as multiple-input multiple-output (MIMO) detection, which can be formulated as a complex quadratic programming problem subject to unit-modulus and discrete argument constraints. Various semidefinite-relaxation-based (SDR-based) algorithms have been proposed to solve the problem in the literature. In this paper, we first show that conventional SDR is generally not tight for the problem. Then, we propose a new and enhanced SDR and show its tightness under an easily checkable condition, which essentially requires the level of the noise to be below a certain threshold. The above results have answered an open question posed by So in [Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'10), SIAM, Philadelphia, PA, 2011, pp. 698--711]. Numerical simulation results show that our proposed SDR significantly outperforms the conventional SDR in terms of the relaxation gap. [ABSTRACT FROM AUTHOR]

We study the iteration complexity of the optimistic gradient descent-ascent (OGDA) method and the extragradient (EG) method for finding a saddle point of a convex-concave unconstrained min-max problem. To do so, we first show that both OGDA and EG can be interpreted as approximate variants of the proximal point method. This is similar to the approach taken in (A. Nemirovski (2004), SIAM J. Optim., 15, pp. 229-251) which analyzes EG as an approximation of the "conceptual mirror prox." In this paper, we highlight how gradients used in OGDA and EG try to approximate the gradient of the proximal point method. We then exploit this interpretation to show that both algorithms produce iterates that remain within a bounded set. We further show that the primal-dual gap of the averaged iterates generated by both of these algorithms converge with a rate of O(1/k). Our theoretical analysis is of interest as it provides the first convergence rate estimate for OGDA in the general convex-concave setting. Moreover, it provides a simple convergence analysis for the EG algorithm in terms of function value without using a compactness assumption. [ABSTRACT FROM AUTHOR]

A convex optimization problem in conic form involves minimizing a linear functional over the intersection of a convex cone and an affine subspace. In some cases, it is possible to replace a conic formulation using a certain cone, with a “lifted” conic formulation using another cone that is higher-dimensional, but simpler, in some sense. One situation in which this can be computationally advantageous is when the higher-dimensional cone is a Cartesian product of many “low-complexity” cones, such as second-order cones, or small positive semidefinite cones. This paper studies obstructions to a convex cone having a lifted representation with such a product structure. The main result says that whenever a convex cone has a certain neighborliness property, then it does not have a lifted representation using a finite product of cones, each of which has only short chains of faces. This is a generalization of recent work of Averkov [SIAM J. Appl. Alg. Geom., 3 (2019), pp. 128–151], which considers only lifted representations using products of positive semidefinite cones of bounded size. Among the consequences of the main result is that various cones related to nonnegative polynomials do not have lifted representations using products of “low-complexity” cones, such as smooth cones, the exponential cone, and cones defined by hyperbolic polynomials of low degree. [ABSTRACT FROM AUTHOR]

The classical convergence analysis of quasi-Newton methods assumes that function and gradient evaluations are exact. In this paper, we consider the case when there are (bounded) errors in both computations and establish conditions under which a slight modification of the BFGS algorithm with an Armijo–Wolfe line search converges to a neighborhood of the solution that is determined by the size of the errors. One of our results is an extension of the analysis presented in [R. H. Byrd and J. Nocedal, SIAM J. Numer. Anal., 26 (1989), pp. 727–739], which establishes that, for strongly convex functions, a fraction of the BFGS iterates are good iterates. We present numerical results illustrating the performance of the new BFGS method in the presence of noise. [ABSTRACT FROM AUTHOR]

This paper presents a finite-difference quasi-Newton method for the minimization of noisy functions. The method takes advantage of the scalability and power of BFGS updating, and employs an adaptive procedure for choosing the differencing interval h based on the noise estimation techniques of Hamming [Introduction to Applied Numerical Analysis, Courier Corporation, North Chelmsford, MA, 2012] and Mor\'e and Wild [SIAM J. Sci. Comput., 33 (2011), pp. 1292--1314]. This noise estimation procedure and the selection of h are inexpensive but not always accurate, and to prevent failures the algorithm incorporates a recovery mechanism that takes appropriate action in the case when the line-search procedure is unable to produce an acceptable point. A novel convergence analysis is presented that considers the effect of a noisy line-search procedure. Numerical experiments comparing the method to a function interpolating trust-region method are presented. [ABSTRACT FROM AUTHOR]

In this paper we present the analytic solutions of the following constrained matrix determinant and trace minimization and maximization problems: minU1,...,Um∈ Un Q | det(cIn ± ∏ m j=1 AjUj )|, maxU1,...,Um∈Un | det(cIn ± ∏mj=1 AjUj )| and minU1,...,Um∈ Un | tr(cIn ± ∏ m j=1 AjUj )|, maxU1,...,Um∈Un | tr(cIn ± ∏mj=1 AjUj )|, where c ∈ ℝ, A1, . . ., Am are n × n complex matrices, In is the n × n identity matrix, Un is the set of n × n unitary matrices, and det(·) and tr(·) denote the matrix determinant function and the trace function, respectively. The given results improve on the corresponding ones in Marshall, Olkin, and Arnold [Inequalities: Theory of Majorization and Its Applications, Springer, New York, 2009], Lu [Acta Math. Sinica, 13 (1963), pp. 49–62], and Sun [SIAM J. Matrix Anal. Appl., 20 (1983), pp. 611–625]. Some theoretical and practical applications are presented. In particular, some examples of applications to test signals of mechanical systems and aero engine fault diagnosis are given to show the efficiency of the proposed theoretical results. [ABSTRACT FROM AUTHOR]