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I have the following LVM partition:

# df -h

 Filesystem Size Used Avail Use% Mounted on
 /dev/mapper/data-root 5.3G 4.2G 864M 84% /
 ...
 /dev/mapper/data-install 6.8G 5.7G 783M 89% /install

# vgs
 VG #PV #LV #SN Attr VSize VFree
 data 1 2 0 wz--n- <12.50g 0

# vgdisplay
 --- Volume group ---
 VG Name data
 System ID
 Format lvm2
 Metadata Areas 1
 Metadata Sequence No 3
 VG Access read/write
 VG Status resizable
 MAX LV 0
 Cur LV 2
 Open LV 2
 Max PV 0
 Cur PV 1
 Act PV 1
 VG Size <12.50 GiB
 PE Size 4.00 MiB
 Total PE 3199
 Alloc PE / Size 3199 / <12.50 GiB
 Free PE / Size 0 / 0
 VG UUID jSACsC-wGwy-ki0w-nSCn-kxMy-bLOV-dVN68E


# pvs
 PV VG Fmt Attr PSize PFree
 /dev/sda2 data lvm2 a-- <12.50g 0

# pvdisplay
 --- Physical volume ---
 PV Name /dev/sda2
 VG Name data
 PV Size <12.50 GiB / not usable 3.00 MiB
 Allocatable yes (but full)
 PE Size 4.00 MiB
 Total PE 3199
 Free PE 0
 Allocated PE 3199
 PV UUID CPggVR-sUXw-4gfw-lb2v-cq0j-M5ey-Ct7td3

# lvs
 LV VG Attr LSize Pool Origin Data% Meta% Move Log Cpy%Sync Convert
 install data -wi-ao---- 7.00g
 root data -wi-ao---- <5.50g


# lvdisplay
 --- Logical volume ---
 LV Path /dev/data/install
 LV Name install
 VG Name data
 LV UUID ZXQmp5-L17b-eQyS-g3An-msk7-Jqso-MTaTIw
 LV Write Access read/write
 LV Creation host, time localhost, 2018-09-28 14:42:43 +0000
 LV Status available
 open 1
 LV Size 7.00 GiB
 Current LE 1792
 Segments 1
 Allocation inherit
 Read ahead sectors aut
 - currently set to 8192
 Block device 253:1



 --- Logical volume ---
 LV Path /dev/data/root
 LV Name root
 VG Name data
 LV UUID i3eMbC-rcm1-H2Gm-JJeF-jRxO-nL9h-toHSHQ
 LV Write Access read/write
 LV Creation host, time localhost, 2018-09-28 14:42:55 +0000
 LV Status available
 # open 1
 LV Size <5.50 GiB
 Current LE 1407
 Segments 1
 Allocation inherit
 Read ahead sectors auto
 - currently set to 8192
 Block device 253:0

I am removing one of the logical volumes:

# umount /install
# lvremove /dev/data/install
# sed -i 'N;//dev/mapper/data-install/s#/dev/mapper/data-install.*n##' /etc/fstab

After the removal:

# df -h
 Filesystem Size Used Avail Use% Mounted on
 /dev/mapper/data-root 5.3G 4.2G 867M 84% /
 ...


# vgs
 VG #PV #LV #SN Attr VSize VFre
 data 1 1 0 wz--n- <12.50g 7.00g


# vgdisplay
 --- Volume group ---
 VG Name data
 System ID
 Format lvm2
 Metadata Areas 1
 Metadata Sequence No 4
 VG Access read/write
 VG Status resizable
 MAX LV 0
 Cur LV 1
 Open LV 1
 Max PV 0
 Cur PV 1
 Act PV 1
 VG Size <12.50 GiB
 PE Size 4.00 MiB
 Total PE 3199
 Alloc PE / Size 1407 / <5.50 GiB
 Free PE / Size 1792 / 7.00 GiB
 VG UUID jSACsC-wGwy-ki0w-nSCn-kxMy-bLOV-dVN68E


# pvs
 PV VG Fmt Attr PSize PFree
 /dev/sda2 data lvm2 a-- <12.50g 7.00g


# pvdisplay
 --- Physical volume ---
 PV Name /dev/sda2
 VG Name data
 PV Size <12.50 GiB / not usable 3.00 MiB
 Allocatable yes
 PE Size 4.00 MiB
 Total PE 3199
 Free PE 1792
 Allocated PE 1407
 PV UUID CPggVR-sUXw-4gfw-lb2v-cq0j-M5ey-Ct7td3


# lvs
 LV VG Attr LSize Pool Origin Data% Meta% Move Log Cpy%Sync Convert
 root data -wi-ao---- <5.50g


# lvdisplay
 --- Logical volume ---
 LV Path /dev/data/root
 LV Name root
 VG Name data
 LV UUID i3eMbC-rcm1-H2Gm-JJeF-jRxO-nL9h-toHSHQ
 LV Write Access read/write
 LV Creation host, time localhost, 2018-09-28 14:42:55 +0000
 LV Status available
 # open 1
 LV Size <5.50 GiB
 Current LE 1407
 Segments 1
 Allocation inherit
 Read ahead sectors auto
 - currently set to 8192
 Block device 253:0

How can I shrink the physical volume, volume group and finally the partition altogether by what amounts to the free space after removing the logical volume? Preferably that should be done specifying relative values (to pvresize?) or something similar to lvextend -l 100%FREE <lvolume>. Also I would like to accomplish this using the command line alone (no GUI). It is also worth noting that the root/system logical volume is active and in use at all times.

Well, pvresize doesn't have an option for relative values like lvextend does, so you need to specify the exact size you want. Here's what you can do:

1. Shrink physical volume: pvresize --setphysicalvolumesize 5.5G /dev/sda2
   


2. Shrink the partition with the tool of your choice. If you want to script it you can use parted. Make the partition a little bit bigger than 5.5G to account for any differences in how the size in bytes is calculated, or ensure you're using the same unit in both tools.


3. If the units differed between the tools, expand the physical volume to consume the entire partition: pvresize /dev/sda2