Regularity and well-posedness of a dual program for convex best C 1-spline interpolation

Qi, Houduo; Yang, Xiaoqi

doi:10.1007/s10589-007-9027-y

Cited by 4 publications

(3 citation statements)

References 20 publications

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“…Then, using a convex cubic spline would give a better approximation of the original function, e.g. [29]; for a general reference on splines, see [10]. However, for our portfolio optimization problem, transaction cost functions are generally nonsmooth.…”

Section: Splinesmentioning

confidence: 99%

Large scale portfolio optimization with piecewise linear transaction costs

Potaptchik

Tunçel

Wolkowicz

2008

Optimization Methods and Software

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We consider the fundamental problem of computing an optimal portfolio based on a quadratic mean-variance model for the objective function and a given polyhedral representation of the constraints. The main departure from the classical quadratic programming formulation is the inclusion in the objective function of piecewise linear, separable functions representing the transaction costs. We handle the nonsmoothness in the objective function by using spline approximations. The problem is first solved approximately using a primal-dual interior-point method applied to the smoothed problem. Then, we crossover to an active set method applied to the original nonsmooth problem to attain a high accuracy solution. Our numerical tests show that we can solve large scale problems efficiently and accurately.

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Section: Splinesmentioning

confidence: 99%

Large scale portfolio optimization with piecewise linear transaction costs

Potaptchik

Tunçel

Wolkowicz

2008

Optimization Methods and Software

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show abstract

“…To avoid strong prior assumptions on the functional form, one can also use a non-parametric approach to perform the function estimation. Generally, the nonparametric estimation is based on a given collection of primitive functions, such as local polynomial [21], trigonometric series, spline estimator [9,27] and kernel-type estimator [3]. However, such an approach may face some difficulties such as imposing the convexity constraint and choosing appropriate smoothing parameters (e.g.…”

Section: Introductionmentioning

confidence: 99%

Efficient algorithms for multivariate shape-constrained convex regression problems

Lin¹,

Sun²,

Toh³

2020

Preprint

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Shape-constrained convex regression problem deals with fitting a convex function to the observed data, where additional constraints are imposed, such as component-wise monotonicity and uniform Lipschitz continuity. This paper provides a comprehensive mechanism for computing the least squares estimator of a multivariate shape-constrained convex regression function in R d . We prove that the least squares estimator is computable via solving a constrained convex quadratic programming (QP) problem with (n + 1)d variables and at least n(n − 1) linear inequality constraints, where n is the number of data points. For solving the generally very large-scale convex QP, we design two efficient algorithms, one is the symmetric Gauss-Seidel based alternating direction method of multipliers (sGS-ADMM), and the other is the proximal augmented Lagrangian method (pALM) with the subproblems solved by the semismooth Newton method (SSN). Comprehensive numerical experiments, including those in the pricing of basket options and estimation of production functions in economics, demonstrate that both of our proposed algorithms outperform the state-of-the-art algorithm. The pALM is more efficient than the sGS-ADMM but the latter has the advantage of being simpler to implement.

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“…To avoid strong prior assumptions on the functional form, one can also use a non-parametric approach to perform the function estimation. Generally, the nonparametric estimation is based on a given collection of primitive functions, such as local polynomial [29], trigonometric series, spline estimator [15,36] and kernel-type estimator [3]. However, such an approach may face some difficulties in imposing the convexity constraint and choosing appropriate smoothing parameters (e.g.…”

Section: Introductionmentioning

confidence: 99%

An augmented Lagrangian method with constraint generation for shape-constrained convex regression problems

Lin¹,

Sun²,

Toh³

2020

Preprint

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Shape-constrained convex regression problem deals with fitting a convex function to the observed data, where additional constraints are imposed, such as component-wise monotonicity and uniform Lipschitz continuity. This paper provides a unified framework for computing the least squares estimator of a multivariate shape-constrained convex regression function in R d . We prove that the least squares estimator is computable via solving a constrained convex quadratic programming (QP) problem with (n + 1)d variables, n(n − 1) linear inequality constraints and n possibly non-polyhedral inequality constraints, where n is the number of data points. To efficiently solve the generally very large-scale convex QP, we design a proximal augmented Lagrangian method (pALM) whose subproblems are solved by the semismooth Newton method (SSN). To further accelerate the computation when n is huge, we design a practical implementation of the constraint generation method such that each reduced problem is efficiently solved by our proposed pALM. Comprehensive numerical experiments, including those in the pricing of basket options and estimation of production functions in economics, demonstrate that our proposed pALM outperforms the state-of-the-art algorithms, and the proposed acceleration technique further shortens the computation time by a large margin.

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Regularity and well-posedness of a dual program for convex best C 1-spline interpolation

Cited by 4 publications

References 20 publications

Large scale portfolio optimization with piecewise linear transaction costs

Large scale portfolio optimization with piecewise linear transaction costs

Efficient algorithms for multivariate shape-constrained convex regression problems

An augmented Lagrangian method with constraint generation for shape-constrained convex regression problems

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