An illumination problem with tradeoff between coverage of a dataset and aperture angle of a conic light beam

Annabella Astorino; Manlio Gaudioso; Alberto Seeger

doi:10.1007/s11081-015-9293-y

Optimization and Engineering

September 2016, Volume 17, Issue 3, pp 557–575 | Cite as

An illumination problem with tradeoff between coverage of a dataset and aperture angle of a conic light beam

Authors
Authors and affiliations

Annabella Astorino
Manlio GaudiosoEmail author
Alberto Seeger

Article

First Online: 04 November 2015

Received: 12 June 2014

Revised: 18 March 2015

Accepted: 09 October 2015

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Abstract

Let $\{a_i:i\in I\}$ be a finite set in ${\mathbb{R}}^n$. The illumination problem addressed in this work concerns the optimal location and orientation of a conic light beam

$$ R\big (z,y,s\big )= \left\{ x \in {\mathbb{R}}^n : s\,\Vert x-z\Vert - \langle y, x-z\rangle \le 0\right\} .$$

The aperture angle $\vartheta = 2\arccos s$ of the conic light beam is a decreasing function of the sharpness coefficient $ s\in [0,1]$. The problem at hand is to select an apex z in a prescribed compact region $Z\subseteq {\mathbb{R}}^n$ and a unit vector $y\in {\mathbb{R}}^n$ so that the conic light beam R(z, y, s) fulfils two conflicting requirements: it captures as many points $a_i$ as possible and, at the same time, it has a sharpness coefficient s as large as possible.

Keywords

Illumination problem Conic light beam Aperture angle Sharpness coefficient Nonsmooth optimization

Mathematics Subject Classification

90C25 90C26 90C40

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Appendix: The NCVX subroutine

The NCVX (NonConVeX minimization) subroutine implements the bundle type method, cf. Fuduli et al. (2004), for finding a local minimum of a locally Lipschitz function $f:\mathbb {R}^n \rightarrow \mathbb {R}$ under no convexity nor smoothness assumptions. Bundle methods have been initially conceived for minimizing convex nondifferentiable functions, cf. Hiriart-Urruty and Lemaréchal (1993). They are based on the construction of a polyhedral type function approximating from below the objective function f. This approximation becomes more and more accurate as the algorithm proceeds.

NCVX is an iterative method which operates as follows. Let $y \in \mathbb {R}^n$ be the estimate of the minimum at current iteration and assume also that a bundle

$$\begin{aligned} \mathcal B=\{(x_i,f(x_i),g_i): i \in I\} \end{aligned}$$

of information is available. The bundle $\mathcal B$ is a set of triplets indexed by I, with $x_i \in \mathbb {R}^n$ and $g_i\in \partial f(x_i)$. Recall that $\partial f(x)$ stands for the Clarke subdifferential of f at x. For each $i \in I$, one defines the linearization error evaluated at y as the number

$$\begin{aligned} \alpha _i = f(y)-[f(x_i)+\langle g_i,y-x_i\rangle ]. \end{aligned}$$

Since f is not convex, the linearization error $\alpha _i$ is not necessarily nonnegative. For this reason, one divides the index set I into $I_+ =\{i: \alpha _i \ge 0 \}$ and $ I_{-} = \{i:\alpha _i < 0 \}$, and then one introduces the polyhedral type functions

$$\begin{aligned} \hat{f}_+(x)= & {} \max _{i\in I_+} \,\{f(x_i)+\langle g_i, x-x_i\rangle \},\\ \hat{f}_-(x)= & {} \min _{i\in I_-}\,\{f(x_i)+\langle g_i,x-x_i\rangle \}, \end{aligned}$$

which are convex and concave, respectively. These functions can be expressed in terms of the linearization errors and of the displacement d from y, defined as $d=x-y$. One gets

$$\begin{aligned} \Delta _+(d)= & {} f(y)+\,\max _{i\in I_+} \,\{\langle g_i,d\rangle -\alpha _i\},\\ \Delta _-(d)= & {} f(y)+\,\min _{i\in I_-}\, \{\langle g_i,d\rangle -\alpha _i\}. \end{aligned}$$

Observing that $\Delta _+(0)<\Delta _-(0)$, a tentative displacement $d^+$ from y is computed as the optimal solution of an auxiliary problem

$$\begin{aligned} \begin{array}{lll} \text{ minimize } \;\; \gamma \,\Delta _+(d) + \frac{1}{2}\Vert d\Vert ^2 \\ d\in \mathbb {R}^n \\ \Delta _+(d) \le \Delta _-(d) \end{array} \end{aligned}$$

whose objective function is the weighted sum, through the positive parameter $\gamma $, of the model $\Delta _+(d)$ and of the proximity term $(1/2)\Vert d\Vert ^2$. The constraint can be interpreted as a sort of trust region argument. The auxiliary problem can be rewritten as a convex quadratic problem.

As a new tentative iterate $y^+$ one considers $y+d^+$. At such point an appropriate decrease condition on f is tested and, whenever it is fulfilled, $y^+$ becomes the new iterate, otherwise the algorithm remains blocked at y, but a new triplet, referred to point $y^+$, is inserted into the bundle, thus enriching at least one of the two model functions $\Delta _+$ and $\Delta _-$. It is possible to prove finite termination of the algorithm at a point satisfying an approximate stationarity condition.

The original version of the algorithm has been modified to cope with a set of linear constraints by simply embedding them into the auxiliary problem, thus ensuring that the solution $y^+$ is feasible for the original problem.

References

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Authors and Affiliations

Annabella Astorino
- 1
Manlio Gaudioso
- 2
Email author
Alberto Seeger
- 3

1.Istituto di Calcolo e Reti ad Alte Prestazioni C.N.R., c/o Dipartimento di Elettronica Informatica e SistemisticaUniversità delle CalabriaRendeItaly
2.Dipartimento di Elettronica Informatica e SistemisticaUniversità delle CalabriaRendeItaly
3.Department of MathematicsUniversity of AvignonAvignonFrance

An illumination problem with tradeoff between coverage of a dataset and aperture angle of a conic light beam

Abstract

Keywords

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References

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