Calmness modulus of fully perturbed linear programs

Abstract

This paper provides operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping in linear optimization under uniqueness of nominal optimal solutions. Our analysis is developed in two different parametric settings. First, in the framework of canonical perturbations (i.e., perturbations of the objective function and the right-hand-side of the constraints), the paper provides a computationally tractable formula for the calmness modulus, which goes beyond some preliminary results of the literature. Second, in the framework of full perturbations (perturbations of all coefficients), after characterizing the calmness property for the optimal set mapping, the paper provides an operative upper bound for the corresponding calmness modulus, as well as some illustrative examples. We provide two applications related to algorithms traced out from the literature: the first one to a descent method in LP, and the second to a regularization method for linear programs with complementarity constraints.

Appendix: Geometrical perturbation ideas for improved bounds

The primary goal of this section is to provide a sketch of the technical details ensuring that inequality (13) is strict in Example 4.1. The underlying idea is that the norm of $\overline{c}$ is too large in this example to let the strategy followed in Example 4.2 work in Example 4.1 (see Remark 6.1 below). The question of investigating in general to what extent the norm of $\overline{c}$ affects $\mathrm {clm}{\mathcal {S}}\left( \left( \overline{c}, \overline{a},\overline{b}\right) ,\overline{x}\right)$ is left to future research.

Now let us go back to problem (16) and the subsequent table in Example 4.1. Let us write $\mathrm {clm}{\mathcal {S}}\left( \overline{p},\overline{x}\right)$ in the form (14) and assume w.l.o.g. that the associated sequence $\{D^{r}\}_{r\in \mathbb {N}}$ is constant, say $D^{r}=\widehat{D}\in \mathcal {K}_{\overline{p}}\left( \overline{x}\right)$ for all r, according to the lines after (14) in the proof of Theorem 4.2. Next we are going to prove (17).

Looking at the end of the proof of Theorem 4.2 $\left( i\right)$, we realize that $\mathrm {clm}{\mathcal {S}}\left( \overline{p},\overline{x}\right) \le \mathrm {clm}\mathcal {L}_{\widehat{D} }\left( \left( \overline{a},\overline{b},-\overline{a}_{\widehat{D}},- \overline{b}_{\widehat{D}}\right) ,\overline{x}\right)$, so that our claim (17) holds automatically if $\widehat{D}$ equals either $\{1,2\}$ or $\{2,3\}$ (see the table of Example 4.1).

Let us consider now the case $\widehat{D}=\{3\}$ and set $\varepsilon _{r}\,{:=}\,\left\| p^{r}-\overline{p}\right\|$ for all r, with $p^{r}=\left( c^{r},a^{r},b^{r}\right)$. Next, we relax the constraints determining $\mathcal {L}_{\{3\}}\left( a^{r},b^{r},-a_{\{3\}}^{r},-b_{\{3\}}^{r}\right)$ (which contains $x^{r}$) around $x^{r}$ in an appropriate way. Specifically, after some calculations one can check that $x^{r}$ is a solution of the following system:

$$\begin{aligned} \left. \begin{array}{rl} \left( \overline{a}_{t}-\alpha _{r}\overline{x}/\left\| \overline{x} \right\| \right) ^{\prime }x &{} \le \overline{b}_{t}+\alpha _{r},\text { for }t=1,2,3,\\ \left( 1-\alpha _{r}\right) x_{1}-\dfrac{\alpha _{r}\left( 1+\alpha _{r}\right) }{10}x_{2} &{} \le 2+\alpha _{r}, \end{array} \right\} \end{aligned}$$

(25)

with $\alpha _{r}\,{:=}\,\dfrac{\left\| \overline{x}\right\| +\beta \varepsilon _{r}}{\left\| \overline{x}\right\| -\beta \varepsilon _{r}} \varepsilon _{r}$, for a scalar $\beta >5+\sqrt{5}$ arbitrarily chosen, and r being assumed to be large enough to ensure $\left\| \overline{x} \right\| -\beta \varepsilon _{r}>0$ and $\left\| x^{r}-\overline{x} \right\| <\beta \varepsilon _{r}$. The last inequality of (25) is inspired by the fact that $\varepsilon _{r}\ge \left\| c^{r}-\overline{c}\right\| \ge d\left( \overline{c},{\mathbb {R}} a_{3}^{r}\right)$.

The reader can check via a routinary computation that, if $\widetilde{x}^{r}$ stands for the furthest solution of (25) with respect to $\overline{x}$, then one has

$$\begin{aligned} \left\| \widetilde{x}^{r}-\overline{x}\right\| \approx \frac{\sqrt{820 \sqrt{5}+3142}}{10}\alpha _{r} \end{aligned}$$

(26)

(i.e., $\lim _{r\rightarrow \infty }\left\| \widetilde{x}^{r}-\overline{x} \right\| /\alpha _{r}=(1/10)\sqrt{820\sqrt{5}+3142}\approx 7.0538$), which together with the obvious fact that $\left\| x^{r}-\overline{x} \right\| \le \left\| \widetilde{x}^{r}-\overline{x}\right\|$—since $x^{r}$ is a solution of (25)—yields (17) by taking into account that $\alpha _{r}\approx \varepsilon _{r}$ as $r\rightarrow \infty$.

The remaining case $\widehat{D}=\{1,3\}$ is very similar to $\widehat{D} =\{3\}$. Indeed (25) still holds at $x=x^{r}$ in the subcase $a_{32}^{r}>0$, with $a_{32}^{r}$ standing for the second coordinate of $a_{3}^{r}$. The subcase $a_{32}^{r}\le 0$ is also very similar, but replacing the fourth inequality of (25) with $\left( 1-\alpha _{r}\right) x_{1}\le 2+\alpha _{r}$. In this subcase the corresponding counterpart of (26) reads as

$$\begin{aligned} \left\| \widetilde{x}^{r}-\overline{x}\right\| \approx \sqrt{8\sqrt{5} +30}~\alpha _{r} \end{aligned}$$

with $\sqrt{8\sqrt{5}+30}\approx 6.9202$, leading again to (17).

Remark 6.1

Coming back to example 4.1, if we perturbed there the constraint system as in (18), then the minimum perturbation of $\overline{c}=\left( 10,0\right) ^{\prime }$ (yielding a perturbed $c^{r}$) making point $x^{r}$ in (19) belong to ${\mathcal {S}}\left( p^{r}\right)$ would satisfy

$$\begin{aligned} \left\| c^{r}-\overline{c}\right\| =d\left( -\overline{c},\mathrm {cone} \{a_{1}^{r},a_{3}^{r}\}\right) \approx 2\sqrt{5}/r, \end{aligned}$$

while $\left\| \left( a^{r},b^{r}\right) -\left( \overline{a}, \overline{b}\right) \right\| =1/r$ and, accordingly we would obtain the smaller ratio $\left\| x^{r}-\overline{x}\right\| /\left\| p^{r}-\overline{p}\right\| \approx \left( 5+\sqrt{5}\right) /\left( 2\sqrt{5}\right) \approx 1.618$.