## Thursday, February 18, 2021

### Restarting the MATE Panel

I know that Cinnamon is the trendy choice for desktop environment with Linux Mint,  but ever since an unfortunate misadventure with video drivers I have been using the somewhat more stable and somewhat faster MATE (pronounced Ma-Tay, per their home page) environment. Overall I am very happy with it. There are, however, occasional hiccups with the panel, the bar along one edge of the display (bottom in my case) containing launchers, tabs for open applications and general what-not. Occasionally, for reasons beyond me ken, some of the icons will be screwed up, duplicated, or duplicated and screwed-up. This morning, for instance, I found not one but three iterations of the audio output icon (looks like a loud speaker, used to set audio preferences). The first icon had the normal appearance, meaning audio was enabled, while the second and third were indicating that audio was muted. (Despite the 2-1 vote against, audio was in fact enabled.)

Glitches like that do not render the system unusable, but they are annoying. So I dug around a bit and discovered that the system command mate-panel. Run from a shell script or a launcher, the command mate-panel --replace seems to do the trick of restarting the panel (and hopefully fixing the glitch that made you restart it).  If you run the command from a terminal, be sure to push it to the background using an ampersand (mate-panel --replace &). Otherwise, it will tie up the terminal session, and when you break out of it (probably via Ctrl-C) the panel will rather unhelpfully evaporate.

## Monday, February 15, 2021

### Lagrangean Relaxation for an Assignment Problem

A question on OR Stack Exchange asked about solving an assignment problem "heuristically but to optimality". The problem formulation (in which I stick as closely as possible to the notation in the original post, but substitute symbols for two numeric parameters) is as follows:

\begin{align*} \max_{d_{u,c}} & \sum_{u=1}^{U}\sum_{c=1}^{C}\omega_{u,c}d_{u,c}\\ \text{s.t. } & \sum_{c=1}^{C}d_{u,c}\le C_{\max}\ \forall u\in\left\{ 1,\dots,U\right\} \\ & \sum_{c=1}^{C}d_{u,c}\ge1\ \forall u\in\left\{ 1,\dots,U\right\} \\ & \sum_{u=1}^{U}d_{u,c}\le U_{\max}\ \forall c\in\left\{ 1,\dots,C\right\} \\ & d_{u,c}\in\left\{ 0,1\right\} \ \forall u,c. \end{align*} Here $d_{u,c}$ is a binary variable, representing assignment of "user" $u$ to "service provider" $c$, and everything else is a parameter. Each user must be assigned at least one provider and at most $C_\max$ providers, and each provider can be assigned at most $U_\max$ users. The objective maximizes the aggregate utility of the assignments.

One of the answers to the question asserts that the constraint matrix has the "integrality property", meaning that any basic feasible solution of the LP relaxation will have integer variable values. The recommended solution approach is therefore to solve the LP relaxation, and I agree with that recommendation. (I have not seen a proof that the matrix has the integrality property, but in my experiments the LP solution always was integer-valued.) That said, the author did ask about "heuristic" approaches, which got me wondering if there was a way to solve to optimality without solving an LP (and thus requiring access to an LP solver).

I decided to try Lagrangean relaxation, and it seems to work. In theory, it should work: if the constraint matrix has the integrality property, and the LP relaxation automatically produces an optimal integer-valued solution, then there is no duality gap, so the solution to the Lagrangean problem should be optimal for the original problem. The uncertainty lies more in numerical issues stemming from the solving of the Lagrangean problem.

In what follows, I am going to reverse the middle constraint of the original problem (multiplying both sides by -1) so that all constraints are $\le$ and thus all dual multipliers are nonnegative. If we let $\lambda\ge 0$, $\mu\ge 0$ and $\nu\ge 0$ be the duals for the three sets of constraints, the Lagrangean relaxation is formulated as follows:

$$\min_{\lambda,\mu,\nu\ge0}LR(\lambda,\mu,\nu)=\\\max_{d\in\left\{ 0,1\right\} ^{U\times C}}\left(\sum_{u}\sum_{c}\omega_{u,c}d_{u,c}-\sum_{u}\lambda_{u}\left[\sum_{c}d_{u,c}-C_{\max}\right]\\+\sum_{u}\mu_{u}\left[\sum_{c}d_{u,c}-1\right]-\sum_{c}\nu_{c}\left[\sum_{u}d_{u,c}-U_{\max}\right]\right).$$

We can simplify that a bit:

$$\min_{\lambda,\mu,\nu\ge0}LR(\lambda,\mu,\nu)=\\\max_{d\in\left\{ 0,1\right\} ^{U\times C}}\left(\sum_{u}\sum_{c}\left[\omega_{u,c}-\lambda_{u}+\mu_{u}-\nu_{c}\right]d_{u,c}\\+C_{\max}\sum\lambda_{u}-\sum_{u}\mu_{u}+U_{\max}\sum_{c}\nu_{c}\right).$$

The inner maximization problem is solvable by inspection. Let $\rho_{u,c}= \omega_{u,c}-\lambda_{u}+\mu_{u}-\nu_{c}$. If $\rho_{u,c} > 0$, $d_{u,c}=1$. If $\rho_{u,c} < 0$, $d_{u,c}=0$. If $\rho_{u,c} = 0$, it does not matter (as far as the inner problem goes) what value we give $d_{u,c}$. So we can rewrite the outer (minimization) problem as follows:

$$\min_{\lambda, \mu, \nu \ge 0}LR(\lambda,\mu,\nu)=\\\sum_{u}\sum_{c}\left(\rho_{u,c}\right)^{+}+C_{\max}\sum\lambda_{u}-\sum_{u}\mu_{u}+U_{\max}\sum_{c}\nu_{c}.$$

$LR(\lambda,\mu,\nu)$ is a piecewise-linear function of its arguments, with directional gradients, but is not continuously differentiable. (Things get a bit tricky when you are on a boundary between linear segments, which corresponds to having $\rho_{u,c}=0$ for one or more combinations of $u$ and $c$.)

I coded a sample instance in R and tested both solving the LP relaxation (using CPLEX) and solving the Lagrangean problem, both using a derivative-based method (a version of the BFGS algorithm) and using a couple of derivative-free algorithms (versions of the Nelder-Mead and Hooke-Jeeves [1] algorithms). Importantly, all three algorithms are modified to allow box constraints, so that we can enforce the sign restriction on the multipliers.

You can download my code in the form of an R notebook, containing text, output and the code itself (which can be extracted). In addition to CPLEX, it uses a gaggle of R libraries: magrittr (for convenience); ompr, ompr.roi, ROI and ROI.plugin.cplex for building the LP model and interfacing with CPLEX; and dfoptim for the Nelder-Mead and Hooke-Jeeves algorithms. (The BFGS algorithm comes via the optim() method, part of the built-in stats library.) If you want to play with the code but do not have CPLEX or some of the libraries, you can just delete the lines that load the missing libraries along with the code that uses them.

Based on limited experimentation, I would say that Nelder-Mead did not work well enough to consider, and BFGS did well in some cases but produced somewhat suboptimal results in others. It may be that tweaking some control setting would have helped with the cases where BFGS ran into trouble. Hooke-Jeeves, again in limited testing, consistently matched the LP solution. So if I needed to come up with some hand-coded way to solve the problem without using libraries (and did not want to write my own simplex code), I would seriously consider using Hooke-Jeeves (which I believe is pretty easy to code) on the Lagrangean problem.

[1] Hooke, Robert and Jeeves, T. (1961) "Direct search'' solution of numerical and statistical problems. Journal of the ACM, Vol. 8, No. 2, 212-229.

## Sunday, January 31, 2021

### Solving a Multidimensional NLP via Line Search

Someone posted a nonconvex nonlinear optimization model on OR Stack Exchange and asked for advice about possible reformulations, piecewise linear approximations, using global optimizers, and other things. The model is as follows:\begin{alignat}{1} \max\,\, & q_{1}+q_{2}\\ \mathrm{s.t.} & \sum_{i=1}^{n}p_{i}x_{i}=\sum_{t=0}^{T}\frac{F_{t}}{(1+q_{1})^{t}} &(1)\\ & \sum_{i=1}^{n}p_{i}x_{i}=\sum_{t=0}^{T}\sum_{i=1}^{n}\frac{b_{t,i}x_{i}}{(1+q_{2})^{t}} &(2)\\ & \sum_{i=1}^{n}p_{i}x_{i}=\beta\sum_{t=0}^{T}\frac{F_{t}}{(1+q_{1}+q_{2})^{t}} &(3)\\ & q_{1,}q_{2}\ge0\\ & x\in\left[0,1\right]^{n} \end{alignat} All symbols other than $q_1$, $q_2$ and $x$ are model parameters (or indexes). The author originally had $x$ as binary variables, apparently believing that would facilitate linearization of products, but also expressed interest in the case where $x$ is continuous. I'm going to propose a possible "low-tech" solution procedure for the continuous case. The binary case is probably a bit too tough for me. The author supplied sample data for all parameters except $\beta$, with dimensions $n=3$ and $T=4$ but expressed a desire to solve the model for $n=10,000$ and $T=1,200$ (gulp).

Note that the left-hand sides (LHSes) of the three constraints are identical. Let $h()$ be the function on the right-hand side (RHS) of constraint (1), so that the RHS of (1) is $h(q_1)$. $h()$ is a monotonically decreasing function. The RHS of (3) is $\beta h(q_1 + q_2)$. Since the left sides are equal, we have $$\beta h(q_1 + q_2) = h(q_1) \quad (4)$$which tells us several things. First, $q_2 \ge 0 \implies h(q_1+q_2) \le h(q_1)$, so if $beta<1$ it is impossible to satisfy (4). Second, if $\beta =1$, (4) implies that $q_2 = 0$, which simplifies the problem a bit. Lastly, let's assume $\beta > 1$. For fixed $q_1$ the LHS of (4) is monotonically decreasing in $q_2$, with the LHS greater than the RHS when $q_2 = 0$ and with $$\lim_{q_2\rightarrow \infty} \beta h(q_1+q_2) = \beta F_0.$$ If $\beta F_0 > h(q_1)$, there is no $q_2$ that can balance equation (4), and so the value of $q_1$ is infeasible in the model. If $\beta F_0 < h(q_1)$, then there is exactly one value of $q_2$ for which (4) holds, and we can find it via line search.

Next, suppose that we have a candidate value for $q_1$ and have found the unique corresponding value of $q_2$ by solving (4). We just need to find a vector $x\in [0,1]^n$ that satisfies (1) and (2). Equation (3) will automatically hold if (1) does, given (4). We can find $x$ by solving a linear program that minimizes 0 subject to (1), (2) and the bounds for $x$.

Thus, we have basically turned the problem into a line search on $q_1$. Let's set an arbitrary upper limit of $Q$ for $q_1$ and $q_2$, so that our initial "interval of uncertainty" for $q_1$ is $[0, Q]$. It's entirely possible that neither 0 nor $Q$ is a feasible value for $q_1$, so our first task is to search upward from $0$ until we find a feasible value (call it $Q_\ell$) for $q_1$, then downward from $Q$ until we find a feasible value (call it $Q_h$) for $q_1$. After that, we cross our fingers and hope that all $q_1 \in [Q_\ell,Q_h]$ are feasible. I think this is true, although I do not have a proof. (I'm much less confident that it is true if we require $x$ to be binary.) Since $q_2$ is a function of $q_1$ and the objective function does not contain $x$, we can search $[Q_\ell,Q_h]$ for a local optimum (for instance, by golden section search) and hope that the objective function is unimodal as a function of $q_1$, so that the local optimum is a global optimum. (Again, I do not have proof, although I would not be surprised if it were true.)

I put this to the test with an R script, using the author's example data. The linear programs were solved using CPLEX, with the model expressed via the OMPR package for R and using ROI as the intermediary between OMPR and CPLEX. I first concocted an arbitrary feasible solution and used it to compute $\beta$, so that I would be sure that the problem was feasible with my choice of $\beta$. Using $\beta = 1.01866$ and 100 (arbitrarily chosen) as the initial upper bounds for $q_1$ and $q_2$, my code produced an "optimal" solution of $q_1= 5.450373$, $q_2 = 0.4664311$, $x = (1, 0.1334608, 0)$ with objective value $5.916804$. There is a bit of rounding error involved: the common LHS of (1)-(3) evaluated to 126.6189, while the three RHSes were 126.6186, 126.6188, and 126.6186. (In my graduate student days, our characterization of this would be "good enough for government work".) Excluding loading libraries, the entire process took under three seconds on my desktop computer.

You can access my R code from my web site. It is in the form of an R notebook (with the code embedded), so even if you are not fluent in R, you can at least read the "prose" portions and see some of the nagging details involved.

## Thursday, January 28, 2021

### A Monotonic Assignment Problem

A question posted on Stack Overflow can be translated to an assignment problem with a few "quirks". First, the number of sources ($m$) is less than the number of sinks ($n$), so while every source is assigned to exactly one sink, not every sink is assigned to a source. Second, there are vectors $a\in\mathbb{R}^m$ and $b\in\mathbb{R}^n$ containing weights for each source and sink, and the cost of assigning source $i$ to sink $j$ is $a_i \times b_j$. Finally, there is a monotonicity constraint. If source $i$ is assigned to sink $j$, then source $i+1$ can only be assigned to one of the sinks $j+1,\dots,n$.

Fellow blogger Erwin Kalvelagen posted a MIP model for the problem and explored some approaches to solving it. A key takeaway is that for a randomly generated problem instance with $m=100$ and $n=1,000$, CPLEX needed about half an hour to get a provably optimal solution. After seeing Erwin's post, I did some coding to cook up a network (shortest path) solution in Java. Several people proposed similar and in some cases essentially the same model in comments on Erwin's post. Today, while I was stuck on a Zoom committee call and fighting with various drawing programs to get a legible diagram, Erwin produced a follow-up post showing the network solution (including the diagram I was struggling to produce ... so I'll refer readers to Erwin's post and forget about drawing it here).

The network is a layered digraph (nodes organized in layers, directed arcs from nodes in one layer to nodes in the next layer). It includes two dummy nodes (a start node in layer 0 and a finish node in layer $m+1$). All nodes in layer $i\in \lbrace 1,\dots,m \rbrace$ represent possible sink assignments for source $i$. The cost of an arc entering a node representing sink $j$ in layer $i$ is $a_i \times b_j$, regardless of the source of the arc. All nodes in layer $m$ connect to the finish node via an arc with cost 0. The objective value of any valid assignment is the sum of the arc costs in the path from start to finish corresponding to that assignment, and the optimal solution corresponds to the shortest path from start to finish.

The monotonicity restriction is enforced simply by omitting arcs from any node in layer $i$ to a lower-index node in layer $i+1$. It also allows us to eliminate some nodes. In the first layer after the start node (where we assign source 1), the available sinks are $1,\dots,n-m+1$. The reason sinks $n-m+2,\dots,n$ are unavailable is that assigning source 1 to one of them and enforcing monotonicity would cause us to run out of sinks before we had made an assignment for every source. Similarly, nodes in layer $i>1$ begin with sink $i$ (because the first $i-1$ sinks must have been assigned or skipped in earlier layers) and end with sink $n-m+i$ (to allow enough sinks to cover the remaining $m-i$ nodes).

For the dimensions $m=100$, $n=1000$, the network has 90,102 nodes and 40,230,551 arcs. That may sound like a lot, but my Java code solves it in under four seconds, including the time spent setting up the network. I used the excellent (open-source) algs4 library, and specifically the AcyclicSP class for solving the shortest path problem. Erwin reports even faster solution time for his network model (0.9 seconds, coded in Python), albeit on different hardware. At any rate, he needed about half an hour to solve the MIP model, so the main takeaway is that for this problem the network model is much faster.

If anyone is interested, my Java code is available for download from my Git repository. The main branch contains just the network model, and the only dependency is the algs4 library. There is also a branch named "CPLEX" which contains the MIP model, in case you either want to compare speeds or just confirm that the network model is getting correct results. If you grab that branch, you will also need to have CPLEX installed.

## Friday, January 22, 2021

### Rainbow Parentheses in RStudio

I use the open-source edition of the RStudio IDE for any R coding I do, and I'm a big fan of it. The latest version (1.4.1103) introduced a new feature that was previously only available in alpha and beta versions: rainbow parentheses. I'd never heard the term before, but the meaning turns out to be remarkably simple. When turned on, if you enter an expression with nested parentheses, brackets or braces, RStudio automatically color codes the parentheses (brackets, braces) to make it easier to see matching pairs. This is in addition to the existing feature that highlights the matching delimiter when you put the cursor after a delimiter.

I was geeked to try this out, but when I first installed the latest version and turned it on, I did not see any changes. Eventually I figured out that it was color coding the delimiters, but the differences were too subtle for me to see. (This was with the default Textmate theme for the IDE.) So I hacked a new theme which makes the colors easier to see. I'll go through the steps here.

First, let me point to some documentation. In a blog post, the folks at RStudio explain how to turn rainbow parentheses on, either globally or just for specific files, and near the end tell which CSS classes need to be tweaked to customize the colors (.ace_paren_color_0 to .ace_paren_color_6). A separate document discusses how to create custom themes.

Theme selection in RStudio is done via Tools > Global Options... > Appearance > Editor theme. Since I use the default (Textmate) theme, my first step was to track down that file and make a copy with a new name. On my Linux Mint system, the file is /usr/lib/rstudio/resources/themes/textmate.rstheme. On Windows (and Macs?) the built-in themes will be lurking somewhere else. The customization document linked above alluded to a ~/.R/rstudio/themes directory on Linux and Macs, but that directory did not exist for me. (I created it, and parked my hacked theme file there.) Put the copied file someplace under a new name. I'm not sure whether the file name is significant to RStudio, but better safe than sorry.

Open the copy you made of your preferred theme file in a text editor. The first two lines are comments that define the theme name (as it will appear in the list of editor themes in RStudio) and whether it is a dark theme or not. Change the name to something that won't conflict with existing themes. In my case, the first line was

/* rs-theme-name: Textmate (default) */
which I changed to
/* rs-theme-name: Paul */
(not very clever, but it got the job done).

Now add code at the bottom to define colors for the seven "ace_paren_color" styles. Here's what I used:

.ace_paren_color_0 {
color: #000000
/* black */
}

.ace_paren_color_1 {
color: #ff00ff
/* magenta */
}

.ace_paren_color_2 {
color: #ffff00
/* yellow */
}

.ace_paren_color_3 {
color: #0080ff
/* light blue */
}

.ace_paren_color_4 {
color: #FF0000
/* red */
}

.ace_paren_color_5 {
color: #004f39
/* Spartan green */
}

.ace_paren_color_6 {
color: #0000ff
/* dark blue */
}


Once you have a candidate style to test, go to the editor themes settings and use the Add... button to add it. I had to fight through a lot of complaints from RStudio, and I needed to restart RStudio to get the new theme to appear. In the same dialog, I selected it, and then put it to the test by typing some nonsense with lots of nested parentheses in a file.

There are two things to watch out for if you try to remove a theme (using the Remove button in that dialog). First, you cannot remove the currently selected theme, so you will need to select a different theme and click Apply, then go back and select the theme to remove. Second, if you remove a theme, RStudio will delete the theme file. So be sure you have a backup copy if you think you might want to use it again (or edit it).

One good thing: once you have added your theme, you can edit your theme without having to remove it and then add it back. After saving any changes, you just have to switch to some other theme and then switch back to your theme to see the impact of the changes in your documents.