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This paper analyzes the convergence rates of the {\it Frank-Wolfe } method for solving convex constrained multiobjective optimization. We establish improved convergence rates under different assumptions on the objective function, the feasible set, and the localization of the limit point of the sequence generated by the method. In terms of the objective function values, we firstly show that if the objective function is strongly convex and the limit point of the sequence generated by the method lies in the relative interior of the feasible set, then the algorithm achieves a linear convergence rate. Next, we focus on a special class of problems where the feasible constraint set is $(\alpha,q)$-uniformly convex for some $\alpha >0$ and $q \geq 2$, including, in particular, \(\ell_p\)-balls for all $p>1$. In this context, we prove that the method attains: (i) a rate of $\mathcal{O}(1/k^\frac{q}{q-1})$ when the objective function is strongly convex; and (ii) a linear rate (if $q=2$) or a rate of $\mathcal{O}(1/k^{\frac{q}{q-2}})$ (if $q>2$) under an additional assumption, which always holds if the feasible set does not contain an unconstrained weak Pareto point. We also discuss enhanced convergence rates for the algorithm in terms of an optimality measure. Finally, we provide some simple examples to illustrate the convergence rates and the set of assumptions.

This paper presents and investigates an inexact proximal gradient method for solving composite convex optimization problems characterized by an objective function composed of a sum of a full domain differentiable convex function and a non-differentiable convex function. We introduce an explicit linesearch strategy that requires only a relative inexact solution of the proximal subproblem per iteration. We prove the convergence of the sequence generated by our scheme and establish its iteration complexity, considering both the functional values and a residual associated with first-order stationary solutions. Additionally, we provide numerical experiments to illustrate the practical efficacy of our method.

Clay minerals are important components of sandstone rocks, due to their significant role in petrophysical properties like porosity and permeability. These minerals have a particular impact on Nuclear Magnetic Resonance measurements since the iron inside clays generates internal gradients that impact the transverse relaxation directly. Here, we apply a methodology recently developed to a set of 20 sandstones with diverse clay content and mineralogy to estimate the total clay content. This estimation relies on the effect of internal gradients and restricted diffusion on transverse relaxation. Our analysis revealed a linear correlation between the total clay content and the displacement of the peak of the T$_2$ distribution as a function of $\tau$, which is half the echo time in the CPMG sequence. Based on these measurements, we propose a geochemical rock typing from quantities determined by our measurements, namely total clay content and porosity., Comment: New improved version

The purpose of this technical report is to review the main properties of an accelerated composite gradient (ACG) method commonly referred to as the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA). In addition, we state a version of FISTA for solving both convex and strongly convex composite minimization problems and derive its iteration complexities to generate iterates satisfying various stopping criteria, including one which arises in the course of solving other composite optimization problems via inexact proximal point schemes. This report also discusses different reformulations of the convex version of FISTA and how they relate to other formulations in the literature.

This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving smooth nonconvex composite optimization problems with nonlinear $\cal K$-convex constraints, i.e., the constraints are convex with respect to the order given by a closed convex cone $\cal K$. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a Lagrange multiplier update. Under some mild assumptions, it is shown that NL-IAPIAL generates an approximate stationary solution of the constrained problem in ${\cal O}(\log(1/\rho)/\rho^{3})$ inner iterations, where $\rho>0$ is a given tolerance. Numerical experiments are also given to illustrate the computational efficiency of the proposed method.

This paper establishes the iteration-complexity of an inner accelerated inexact proximal augmented Lagrangian (IAIPAL) method for solving linearly-constrained smooth nonconvex composite optimization problems that is based on the classical augmented Lagrangian (AL) function. More specifically, each IAIPAL iteration consists of inexactly solving a proximal AL subproblem by an accelerated composite gradient (ACG) method followed by a classical Lagrange multiplier update. Under the assumption that the domain of the composite function is bounded and the problem has a Slater point, it is shown that IAIPAL generates an approximate stationary solution in ${\cal O}(\varepsilon^{-5/2}\log^2 \varepsilon^{-1})$ ACG iterations where $\varepsilon>0$ is a tolerance for both stationarity and feasibility. Moreover, the above bound is derived without assuming that the initial point is feasible. Finally, numerical results are presented to demonstrate the strong practical performance of IAIPAL.

This paper proposes and establishes the iteration-complexity of an inexact proximal accelerated augmented Lagrangian (IPAAL) method for solving linearly constrained smooth nonconvex composite optimization problems. Each IPAAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a suitable Lagrange multiplier update. It is shown that IPAAL generates an approximate stationary solution in at most ${\cal O}(\log(1/\rho)/\rho^{3})$ ACG iterations, where $\rho>0$ is the given tolerance. It is also shown that the previous complexity bound can be sharpened to ${\cal O}(\log(1/\rho)/\rho^{2.5})$ under additional mildly stronger assumptions. The above bounds are derived assuming that the initial point is neither feasible nor the domain of the composite term of the objective function is bounded. Some preliminary numerical results are presented to illustrate the performance of the IPAAL method.

This paper proposes an efficient adaptive variant of a quadratic penalty accelerated inexact proximal point (QP-AIPP) method proposed earlier by the authors. Both the QP-AIPP method and its variant solve linearly set constrained nonconvex composite optimization problems using a quadratic penalty approach where the generated penalized subproblems are solved by a variant of the underlying AIPP method. The variant, in turn, solves a given penalized subproblem by generating a sequence of proximal subproblems which are then solved by an accelerated composite gradient algorithm. The main difference between AIPP and its variant is that the proximal subproblems in the former are always convex while the ones in the latter are not necessarily convex due to the fact that their prox parameters are chosen as aggressively as possible so as to improve efficiency. The possibly nonconvex proximal subproblems generated by the AIPP variant are also tentatively solved by a novel adaptive accelerated composite gradient algorithm based on the validity of some key convergence inequalities. As a result, the variant generates a sequence of proximal subproblems where the stepsizes are adaptively changed according to the responses obtained from the calls to the accelerated composite gradient algorithm. Finally, numerical results are given to demonstrate the efficiency of the proposed AIPP and QP-AIPP variants.

This paper proposes a partially inexact alternating direction method of multipliers for computing approximate solution of a linearly constrained convex optimization problem. This method allows its first subproblem to be solved inexactly using a relative approximate criterion, whereas a proximal term is added to its second subproblem in order to simplify it. A stepsize parameter is included in the updating rule of the Lagrangian multiplier to improve its computational performance. Pointwise and ergodic interation-complexity bounds for the proposed method are established. To the best of our knowledge, this is the first time that complexity results for an inexact ADMM with relative error criteria has been analyzed. Some preliminary numerical experiments are reported to illustrate the advantages of the new method.

This paper analyzes the iteration-complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs. More specifically, the objective function is of the form $f + h$ where $f$ is a differentiable function whose gradient is Lipschitz continuous and $h$ is a closed convex function with bounded domain. The method, basically, consists of applying an accelerated inexact proximal point method for solving approximately a sequence of quadratic penalized subproblems associated to the linearly constrained problem. Each subproblem of the proximal point method is in turn approximately solved by an accelerated composite gradient (ACG) method. It is shown that the proposed scheme generates a $\rho$-approximate stationary point in at most ${\cal{O}}(\rho^{-3})$ ACG iterations. Finally, numerical results showing the efficiency of the proposed method are also given.

Since the dawn of quantum theory, coherence was attributed as a key to understand the weirdness of fundamental concepts like the wave-particle duality and the Stern-Gerlach experiment. Recently, based on a resource theory approach, the notion of quantum coherence was revisited and a plethora of coherence quantifiers were proposed. In this work, we address this issue using the language of coherence orders, developed by the NMR community. This allowed us to investigate the role played by different subspaces of the Hilbert-Schmidt space into physical processes and quantum protocols. We found some links between decoherence and each coherence order. Moreover, we propose a sufficient and straightforward criterion to testify the usefulness of a given state for quantum enhanced phase estimation, relying on a minimal set of elements belonging to the density matrix., Comment: 7 + 8 pages, 1 + 6 figures. Close to published version and with full acknowledgments, as coffee is a fundamental part of scientific work

We present a simple $^1$H NMR approach for characterizing intermediate to fast regime molecular motions using $^1$H time-domain NMR at low magnetic field. The method is based on a Goldmann Shen dipolar filter (DF) followed by a Mixed Magic Sandwich Echo (MSE). The dipolar filter suppresses the signals arising from molecular segments presenting sub kHz mobility, so only signals from mobile segments are detected. Thus, the temperature dependence of the signal intensities directly evidences the onset of molecular motions with rates higher than kHz. The DF-MSE signal intensity is described by an analytical function based on the Anderson Weiss theory, from where parameters related to the molecular motion (e.g. correlation times and activation energy) can be estimated when performing experiments as function of the temperature. Furthermore, we propose the use of the Tikhonov regularization for estimating the width of the distribution of correlation times., Comment: 9 pages, 6 figures

This paper establishes the iteration-complexity of a Jacobi-type non-Euclidean proximal alternating direction method of multipliers (ADMM) for solving multi-block linearly constrained nonconvex programs. The subproblems of this ADMM variant can be solved in parallel and hence the method has great potential to solve large scale multi-block linearly constrained nonconvex programs. Moreover, our analysis allows the Lagrange multiplier to be updated with a relaxation parameter in the interval (0, 2).

In this paper, we obtain global $\mathcal{O} (1/ \sqrt{k})$ pointwise and $\mathcal{O} (1/ {k})$ ergodic convergence rates for a variable metric proximal alternating direction method of multipliers(VM-PADMM) for solving linearly constrained convex optimization problems. The VM-PADMM can be seen as a class of ADMM variants, allowing the use of degenerate metrics (defined by noninvertible linear operators). We first propose and study nonasymptotic convergence rates of a variable metric hybrid proximal extragradient (VM-HPE) framework for solving monotone inclusions. Then, the above-mentioned convergence rates for the VM-PADMM are obtained essentially by showing that it falls within the latter framework. To the best of our knowledge, this is the first time that global pointwise (resp. pointwise and ergodic) convergence rates are obtained for the VM-PADMM (resp. VM-HPE framework).

This paper establishes convergence rate bounds for a variant of the proximal alternating direction method of multipliers (ADMM) for solving nonconvex linearly constrained optimization problems. The variant of the proximal ADMM allows the inclusion of an over-relaxation stepsize parameter belonging to the interval $(0,2)$. To the best of our knowledge, all related papers in the literature only consider the case where the over-relaxation parameter lies in the interval $(0,(1+\sqrt{5})/2)$.

In this chapter we review the contributions of Nuclear Magnetic Resonance to the study of quantum correlations, including its capabilities to prepare initial states, generate unitary transformations, and characterize the final state. These are the three main demands to implement quantum information processing in a physical system, which NMR offers, nearly to perfection, though for a small number of qubits. Our main discussion will concern liquid samples at room temperature., Comment: 24 pages, 14 figures. Contribution to "Lectures on general quantum correlations and their applications", edited by Felipe Fanchini, Diogo Soares-Pinto, and Gerardo Adesso. Sorry for being late!

Pointwise and ergodic iteration-complexity results for the proximal alternating direction method of multipliers (ADMM) for any stepsize in(0,(1+\sqrt{5})/2) have been recently established in the literature. In addition to giving alternative proofs of these results, this paper also extends the ergodic iteration-complexity result to include the case in which the stepsize is equal to (1+\sqrt{5})/$. As far as we know, this is the first ergodic iteration-complexity for the stepsize (1+\sqrt{5})/2 obtained in the ADMM literature. These results are obtained by showing that the proximal ADMM is an instance of a non-Euclidean hybrid proximal extragradient framework whose pointwise and ergodic convergence rate are also studied., Comment: arXiv admin note: text overlap with arXiv:1601.01140

In this paper, we consider the problem of solving a constrained system of nonlinear equations. We propose an algorithm based on a combination of the Newton and conditional gradient methods, and establish its local convergence analysis. Our analysis is set up by using a majorant condition technique, allowing us to prove in a unified way convergence results for two large families of nonlinear functions. The first one includes functions whose derivative satisfies a Holder-like condition, and the second one consists of a substantial subclass of analytic functions. Numerical experiments illustrating the applicability of the proposed method are presented, and comparisons with some other methods are discussed.

We consider the quantum processor based on a chain of trapped ions to propose an architecture wherein the motional degrees of freedom of trapped ions (position and momentum) could be exploited as the computational Hilbert space. We adopt a continuous-variables approach to develop a toolbox of quantum operations to manipulate one or two vibrational modes at a time. Together with the intrinsic non-linearity of the qubit degree of freedom, employed to mediate the interaction between modes, arbitrary manipulation and readout of the ionic wave function could be achieved., Comment: 12 pages, 2 figures

This paper describes a regularized variant of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex programs. It is shown that the pointwise iteration-complexity of the new method is better than the corresponding one for the standard ADMM method and that, up to a logarithmic term, is identical to the ergodic iteration-complexity of the latter method. Our analysis is based on first presenting and establishing the pointwise iteration-complexity of a regularized non-Euclidean hybrid proximal extragradient framework whose error condition at each iteration includes both a relative error and a summable error. It is then shown that the new method is a special instance of the latter framework where the sequence of summable errors is identically zero when the ADMM stepsize is less than one or a nontrivial sequence when the stepsize is in the interval [1, (1 +\sqrt{5})/2).