Convex and Nonsmooth Optimization

What Shape is your Conjugate? A Survey of Computational Convex Analysis and its Applications

  • Yves Lucet
    • Categories Applications - OR and Management Sciences, Convex Optimization Tags , , , , , ,

    Computational Convex Analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of Computational Convex Analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolic-numeric algorithms. Our objective is … Read more

    Nonsmooth Optimization for Production Theory

  • Yoshihiro Tanaka
    • Categories Generalized Convexity/Monoticity

    Production theory needs generalizations so that it can incorporate broader class of production functions. A generalized Hotelling’s lemma and a generalized Shephard’s lemma in economic theory, which are established in virtue of nonsmooth analysis under the assumption of upper semicontinuity on production functions. Continuity of factor inputs with respect to a change of the factor … Read more

    Robust Efficient Frontier Analysis with a Separable Uncertainty Model

  • Stephen Boyd
  • Seung-Jean Kim
    • Categories Convex Optimization, Finance and Economics, Robust Optimization Tags , , , , , , ,

    Mean-variance (MV) analysis is often sensitive to model mis-specification or uncertainty, meaning that the MV efficient portfolios constructed with an estimate of the model parameters (i.e., the expected return vector and covariance of asset returns) can give very poor performance for another set of parameters that is similar and statistically hard to distinguish from the … Read more

    Generating set search methods for piecewise smooth problems

  • Claudio Bogani
  • Maria Grazia Gasparo
  • Alessandra Papini
    • Categories Nonsmooth Optimization Tags , , ,

    We consider a direct search approach for solving nonsmooth minimization problems where the objective function is locally Lipschitz continuous and piecewise continuously differentiable on a finite family of polyhedra. A generating set search method is proposed, which is named “structured” because the structure of the set of nondifferentiability near the current iterate is exploited to … Read more

    A Minimax Theorem with Applications to Machine Learning, Signal Processing, and Finance

  • Stephen Boyd
  • Seung-Jean Kim
    • Categories Convex Optimization, Finance and Economics, Robust Optimization Tags , , , ,

    This paper concerns a fractional function of the form $x^Ta/\sqrt{x^TBx}$, where $B$ is positive definite. We consider the game of choosing $x$ from a convex set, to maximize the function, and choosing $(a,B)$ from a convex set, to minimize it. We prove the existence of a saddle point and describe an efficient method, based on … Read more

    Controlling the dose distribution with gEUD-type constraints within the convex IMRT optimization framework

  • Tim Craig
  • Harald Keller
  • Michael Sharpe
  • Tamás Terlaky
  • Yuriy Zinchenko
    • Categories Biomedical Applications, Convex Optimization Tags , , , , , ,

    Radiation therapy is an important modality in treating various cancers. Various treatment planning and delivery technologies have emerged to support Intensity Modulated Radiation Therapy (IMRT), creating significant opportunities to advance this type of treatment. We investigate the possibility of including the dose prescription, specified by the DVH, within the convex optimization framework for inverse IMRT … Read more

    Support Vector Regression for imprecise data

  • Emilio Carrizosa
  • José F. Gordillo
  • Frank Plastria
    • Categories Convex Optimization, Data-Mining, Quadratic Programming Tags , , , , ,

    In this work, a regression problem is studied where the elements of the database are sets with certain geometrical properties. In particular, our model can be applied to handle data affected by some kind of noise or uncertainty and interval-valued data, and databases with missing values as well. The proposed formulation is based on the … Read more

    Global and adaptive scaling in a separable augmented lagrangian algorithm

  • Arnaud Lenoir
  • Philippe Mahey
    • Categories Convex Optimization Tags ,

    In this paper, we analyze the numerical behaviour of a separable Augmented Lagrangian algorithm with multiple scaling parameters, different parameters associated with each dualized coupling constraint as well as with each subproblem. We show that an optimal superlinear rate of convergence can be theoretically attained in the twice differentiable case and propose an adaptive scaling … Read more

    Sparse Reconstruction by Separable Approximation

  • Mario A. T. Figueiredo
  • Stephen Wright
  • Robert Nowak
    • Categories Applications - Science and Engineering, Convex and Nonsmooth Optimization Tags , , ,

    Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution and reconstruction, and compressed sensing (CS) are a few well-known areas in which problems of this type appear. One standard approach is to … Read more

    Regularization methods for semidefinite programming

  • Jérôme Malick
  • Janez Povh
  • Franz Rendl
  • Angelika Wiegele
    • Categories Convex Optimization, Nonlinear Optimization, Semi-definite Programming Tags , , ,

    This paper studies an alternative technique to interior point methods and nonlinear methods for semidefinite programming (SDP). The approach based on classical quadratic regularizations leads to an algorithm, generalizing a recent method called “boundary point method”. We study the theoretical properties of this algorithm and we show that in practice it behaves very well on … Read more