Foundations of Gauge and Perspective Duality

Aravkin, Aleksandr Y.; Burke, James V.; Friedlander, Michael P.; MacPhee, Kellie J.

doi:10.1137/17m1119020

Cited by 16 publications

(21 citation statements)

References 24 publications

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“…When comparing to AVE, the research associated to AVO problems is insufficient and one of these reasons is the difficulty for obtaining feasible solutions of the problems. In fact, their constraints include AVE, which are known to be NP-hard [14].Another optimization problem that is related to AVO was recently investigated by Friedlander et al [8] and Aravkin et al [2]. It is called gauge optimization, which basically consists in an optimization problem with the so-called gauge function.…”

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confidence: 99%

“…However, differently from AVO, this problem does not consider multiple constraints, but only one gauge constraint. In [2,8], the authors showed that the Lagrange dual of gauge optimization problems can be written in a closed-form by using the polar of the gauge functions.In this paper, similarly to [2,8], we introduce a generalized AVO problem, and show that it has a wider practical application comparing to AVO problems. It is also more general than gauge optimization problems, because multiple constraints can be considered here.…”

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confidence: 99%

“…Another optimization problem that is related to AVO was recently investigated by Friedlander et al [8] and Aravkin et al [2]. It is called gauge optimization, which basically consists in an optimization problem with the so-called gauge function.…”

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Duality of nonconvex optimization with positively homogeneous functions

Yamanaka

Yamashita

2018

Comput Optim Appl

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We consider an optimization problem with positively homogeneous functions in its objective and constraint functions. Examples of such positively homogeneous functions include the absolute value function and the p-norm function, where p is a positive real number. The problem, which is not necessarily convex, extends the absolute value optimization proposed in [O. L. Mangasarian, Absolute value programming, Computational Optimization and Applications 36 (2007) pp. [43][44][45][46][47][48][49][50][51][52][53]. In this work, we propose a dual formulation that, differently from the Lagrangian dual approach, has a closed-form and some interesting properties. In particular, we discuss the relation between the Lagrangian duality and the one proposed here, and give some sufficient conditions under which these dual problems coincide. Finally, we show that some well-known problems, e.g., sum of norms optimization and the group Lasso-type optimization problems, can be reformulated as positively homogeneous optimization problems.Recently, the so-called absolute value equations (AVE) and absolute value optimization (AVO) problems have been attracted much attention. The AVE were introduced in 2004 by Rohn [21]. Basically, ifÃ,B are given matrices, andb is a given vector, one should find a vector x that satisfiesÃx +B|x| =b, where |x| is a vector whose i-th entry is the absolute value of the i-th entry of x. It is known that AVE are equivalent to the linear complementarity problems (LCP) [9,16,20], which include many real-world applications. As an extension of AVE, Mangasarian [14] proposed in 2007 the AVO problems, which have the absolute value of variables in their objective and constraint functions. More precisely, the AVO problem considered is given by+K|x| ≥p, whereÃ,B,H,K are given matrices, andc,d,b,p are vectors with appropriate dimensions. Since AVE and LCP are equivalent, the AVO include the mathematical programs with linear complementarity constraints [12], which are one of the formulations of equilibrium problems. As another application of AVO, Yamanaka and Fukushima [26] presented facility location problems. Since 2007, some methods for solving AVE have been presented in the literature. For example, Rohn [22] considered an iterative algorithm using the sign of variables for the case thatÃ andB are square matrices. For more generalÃ andB, Mangasarian [14] provided a method involving successive linearization techniques. Another methods include a concave minimization approach, given by Mangasarian [13], and Newton-type methods, proposed by Caccetta et al. [3], Mangasarian [15], and Zhang and Wei [28]. Some generalizations of AVE were also proposed. For example, Hu et al. [10] considered an AVE involving the absolute value of variables associated to the second-order cones. Miao et al. [18]investigated an AVE with the so-called circular cones. In both papers, quasi-Newton based algorithms were used.As for AVO problems, Yamanaka and Fukushima [26] proposed to use a branch-andbound technique. In the branching procedure, ...

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Duality of nonconvex optimization with positively homogeneous functions

Yamanaka

Yamashita

2018

Comput Optim Appl

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“…Moreover, each atom is a subgradient of σ A . The theoretical basis for this approach is outlined by Friedlander et al [8] and Aravkin et al [9].…”

Section: Dual Atomic Pursuitmentioning

confidence: 99%

Bundle methods for dual atomic pursuit

Fan

Sun

Friedlander

2019

2019 53rd Asilomar Conference on Signals, Systems, and Computers

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The aim of structured optimization is to assemble a solution, using a given set of (possibly uncountably infinite) atoms, to fit a model to data. A two-stage algorithm based on gauge duality and bundle method is proposed. The first stage discovers the optimal atomic support for the primal problem by solving a sequence of approximations of the dual problem using a bundle-type method. The second stage recovers the approximate primal solution using the atoms discovered in the first stage. The overall approach leads to implementable and efficient algorithms for large problems.

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Maximum regularized likelihood estimators: A general prediction theory and applications

Zhuang

Lederer

2018

Stat

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Maximum regularized likelihood estimators (MRLEs) are arguably the most established class of estimators in highdimensional statistics. In this paper, we derive guarantees for MRLEs in the Kullback-Leibler divergence, a general measure of prediction accuracy. We assume only that the densities have a convex parametrization and that the regularization is definite and positive homogenous. The results thus apply to a very large variety of models and estimators, such as tensor regression and graphical models with convex and non-convex regularized methods. A main conclusion is that MRLEs are broadly consistent in prediction-regardless of whether restricted eigenvalues or similar conditions hold.definiteness and positive homogeneity of the regularizers. This makes the results applicable to a variety of parametric and non-parametric models and allows for a broad class of convex and non-convex regularizers.The remainder of this paper is organized as follows. We introduce the framework and the general result in Section 2. We then provide some examples in Section 3. We finally conclude with a brief discussion in Section 4. All proofs are deferred to the Appendices in the supporting information: proofs for the main result to Appendix A, proofs for the examples to Appendix B, and proofs for the bounds of the empirical process terms in Appendix C. In addition, Appendix D contains notation and properties of tensors. Our contributionThe contribution of this work is twofold. First, Theorem 2.1 provides a general prediction guarantee for MRLEs in terms of the Kullback-Leibler loss. Besides being the first assumptionless bound in such a broad setting, the result specializes correctly to known results, such as for lasso, where the corresponding rates have been shown to be optimal up to log-factors. Second, we show that applications of the general theorem to specific examples lead to new guarantees in tensor response regression, generalized linear tensor regression and graphical modelling. The theory thus also establishes new insights into individual cases of MRLEs.

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Foundations of Gauge and Perspective Duality

Cited by 16 publications

References 24 publications

Duality of nonconvex optimization with positively homogeneous functions

Duality of nonconvex optimization with positively homogeneous functions

Bundle methods for dual atomic pursuit

Maximum regularized likelihood estimators: A general prediction theory and applications

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