---
title: "PIQP Solver Interface"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{PIQP Solver Interface}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup, echo = FALSE}
library(piqp)
```

## 1. Introduction

PIQP solves quadratic programs of the form

$$
\begin{aligned}
\min_{x} \quad & \frac{1}{2} x^\top P x + c^\top x \\
\text {s.t.}\quad & Ax=b, \\
& Gx \leq h, \\
& x_{lb} \leq x \leq x_{ub}
\end{aligned}
$$

with primal decision variables $x \in \mathbb{R}^n$, matrices $P\in
\mathbb{S}_+^n$, $A \in \mathbb{R}^{p \times n}$, $G \in \mathbb{R}^{m
\times n}$, and vectors $c \in \mathbb{R}^n$, $b \in \mathbb{R}^p$, $h
\in \mathbb{R}^m$, $x_{lb} \in \mathbb{R}^n$, and $x_{ub} \in
\mathbb{R}^n$.

## 2. The Problem Solver Interface 

Consider:

$$
\begin{aligned}
\min_{x} \quad & \frac{1}{2} x^\top \begin{bmatrix} 6 & 0 \\ 0 & 4 \end{bmatrix} x + \begin{bmatrix} -1 \\ -4 \end{bmatrix}^\top x \\
\text {s.t.}\quad & \begin{bmatrix} 1 & -2 \end{bmatrix} x = 1, \\
& \begin{bmatrix} 1 & -1 \\ 2 & 0 \end{bmatrix} x \leq \begin{bmatrix} 0.2 \\ -1 \end{bmatrix}, \\
& -1 \leq x_1 \leq 1.
\end{aligned}
$$

The data for this problem can be specified as below.

```{r}
P <- matrix(c(6, 0, 0, 4), nrow = 2)
c <- c(-1, -4)
A <- matrix(c(1, -2), nrow = 1)
b <- 1
G <- matrix(c(1, 2, -1, 0), nrow = 2)
h <- c(0.2, -1)
x_lb <- c(-1, -Inf)  ## 2 variables
x_ub <- c(1, Inf)    ## 2 variables
```

The problem can now be solved via a call to `solve_piqp()`.

```{r}
sol <- solve_piqp(P, c, A, b, G, h, x_lb = x_lb, x_ub = x_ub, backend = "auto")
cat(sprintf("(Solution status, description): = (%d, %s)\n",
            sol$status, sol$info$status_desc))
cat(sprintf("Objective: %f, solution: (x1, x2) = (%f, %f)\n", sol$info$primal_obj, sol$x[1], sol$x[2]))
```

`sol` contains many components as `str(sol)` will display but the most important ones are:

- `status` : 1 if all goes well (more below),
- `x` : solution vector
- `y` : dual solution for the equality constraints
- `z` : dual solution for the inequality constraints
- `z_lb` : dual solution of lower bound box constraints
- `z_ub` : dual solution of upper bound box constraints
- `info$status_desc`: a descriptive string of the status
- `info$primal_pobj` : primal objective value
- `info$run_time` : total runtime, if asked for in settings (see below).

One can always construct the descriptive string for the status using:

```{r}
status_description(sol$status)
```

Note that PIQP can handle infinite box constraints well, i.e. when elements of
$x_{lb}$ or $x_{ub}$ are $-\infty$ or $\infty$, respectively. On the
contrary, infinite values in the general inequalities $Gx \leq h = \pm
\infty$ can cause problems, i.e., they are converted internally to
`-1e30` and `1e30`, respectively.


## 3. The Solver Model Object

Users who wish to solve QP problems will mostly use the `solve_piqp()`
function. Behind the scenes, `solve_piqp()` creates a solver object
and calls methods on the object to obtain the solution. The solver
object can be created explitly using `piqp()` and provides more
elaborate facilities for updating problem data. This can be very
efficient when one is solving the same kind of problem over and
over. 

The above problem could be solved using the solver model object thus:

```{r}
model <- piqp(P, c, A, b, G, h, x_lb = x_lb, x_ub = x_ub)
sol2 <- model$solve()
identical(sol, sol2)
```

Indeed, this is exactly what `solve_piqp()` does. But this interface
allows us to update the settings, the bounds, etc. and resolve the
same problem more efficiently.

```{r}
model$update(x_lb = c(0, 0))
sol3 <- model$solve()
cat(sprintf("(Solution status, description): = (%d, %s)\n",
            sol3$status, status_description(sol3$status)))
```

Setting the lower bounds made the problem infeasible.

```{r, error = TRUE}
## Try to give an inappropriate b value
model$update(b = c(5, 2))
```

The error message correctly indicates that `b` has wrong dimensions. 

The methods exposed by the model object can be seen in the
documentation for the object `piqp_model`.

For example, we could query the problem dimensions.

```{r}
model$get_dims()
```

## 3. Dense and Sparse Interfaces

PIQP supports dense and sparse problem formulations. For small and
dense problems the dense interface is preferred since vectorized
instructions and cache locality can be exploited more efficiently, but
for sparse problems the sparse interface and result in significant
speedups.

Either interface can be requested explicitly via the `backend`
parameter which can take on any value among `"dense"`, `"sparse"`, or
`"auto"`, the default. The last value will automatically switch to a
sparse interface if any of the supplied inputs ($A$, $P$, or $G$) is a
sparse matrix; otherwise it uses the dense interface.

```{r}
sparse_sol <- solve_piqp(P, c, A, b, G, h, x_lb, x_ub, backend = "sparse")
str(sparse_sol)
```

## 5. Another Example

Suppose that we want to solve the following 2-dimensional quadratic programming problem:

$$
\begin{array}{ll} \text{minimize} &  3x_1^2 + 2x_2^2 - x_1 - 4x_2\\
\text{subject to} &  -1 \leq x \leq 1, ~ x_1 = 2x_2
\end{array}
$$

Since the solver expects the objective in the form $\frac{1}{2}x^\top
P x + c^\top x$, we define

$$
P = 2 \cdot \begin{bmatrix} 3 & 0 \\ 0 & 2\end{bmatrix}
\mbox{ and }
q = \begin{bmatrix} -1 \\ -4\end{bmatrix}.
$$

We have one equality constraint and box constraints. This leads to the
following straight-forward formulation.

```{r}
P <- matrix(2 * c(3, 0, 0, 2), nrow = 2, ncol = 2)
c <- c(-1, -4)
A <- matrix(c(1, -2), ncol = 2)
b <- 0
x_lb <- rep(-1.0, 2)
x_ub <- rep(1.0, 2)
sol <- solve_piqp(P = P, c = c, A = A, b = b, x_lb = x_lb, x_ub = x_ub)
cat(sprintf("(Solution status, description): = (%d, %s)\n",
            sol$status, sol$info$status_desc))
cat(sprintf("Objective: %f, solution: (x1, x2) = (%f, %f)\n", sol$info$primal_obj, sol$x[1], sol$x[2]))
```

But we can also choose to move the upper box constraints into the inequalities.

```{r}
G <- diag(2)
h <- c(1, 1)
sol <- solve_piqp(P = P, c = c, A = A, b = b, G = G, h = h,
                  x_lb = c(-1, -1), x_ub = c(Inf, Inf))
cat(sprintf("(Solution status, description): = (%d, %s)\n",
            sol$status, sol$info$status_desc))
cat(sprintf("Objective: %f, solution: (x1, x2) = (%f, %f)\n", sol$info$primal_obj, sol$x[1], sol$x[2]))
```

Or we can move both of them into the inequalities. 

```{r}
G <- Matrix::Matrix(c(1, 0, -1, 0, 0, 1, 0, -1), byrow = TRUE,
                    nrow = 4, sparse = TRUE)
h <- rep(1, 4)

sol <- solve_piqp(A = A, b = b, c = c, P = P, G = G, h = h)
cat(sprintf("(Solution status, description): = (%d, %s)\n",
            sol$status, status_description(sol$status)))
cat(sprintf("Objective: %f, solution: (x1, x2) = (%f, %f)\n", sol$info$primal_obj, sol$x[1], sol$x[2]))
```

All of them will yield the same result.


## 5. Solver parameters

PIQP has a number of parameters that control its behavior, including
verbosity, tolerances, etc.; see help on `piqp_settings()`. As an
example, in the last problem, we can reduce the number of iterations.

```{r}
s <- solve_piqp(P = P, c = c, A = A, b = b, G = G, h = h,
          settings = list(max_iter = 3)) ## Reduced number of iterations
cat(sprintf("(Solution status, description): = (%d, %s)\n",
            s$status, s$info$status_desc))
cat(sprintf("Objective: %f, solution: (x1, x2) = (%f, %f)\n", s$info$primal_obj, s$x[1], s$x[2]))
```

Note the different status, which should always be checked in code.