Some points on bears in warfare, as well as an alternative to bears which would likely be more favorable logistics wise. Oh, and apologies for the lengthy answer.
To start with, how well would a bear work out in war?
To start off with, you could armor a bear. A bear that was kept for war would logically be well fed as it grew, and so would be a good size, note, that means it would be about 1.5 metres at the shoulder on all fours, and 3 metres at the shoulder while on its hind legs.
I took a look at a picture of a rearing grizzly, assumed a height of 3 metres at its shoulders, and an average armor thickness of 2.25mm (the thickness of plate armour was 1.5 - 3 mm, thicker in the areas more likely to be hit, thinner in other areas). If you gave a bear of this size armour to cover it's entire torso, the top three quarters of its arms, the top three quarters of its legs, and a kind of collar protruding from the breast plate to protect the throat, you would use just shy of 90kg of steel. A bear of this size can lift up to 500kg, so this is not a problem.
An unarmoured bear is already difficult to injure, and it is important to note that weapons don't cut steel. When facing an opponent armoured in plate, you aim for week points, areas of the armour which need to be flexible, in this case, the armpits, which can be protected with pauldrons or besagews, the elbows, and the backs of the knee's.
This would be difficult to do, your average polearm tips out at a length of 1.8 metres, but only half of that extends out in front of your hands, and a medieval man had an arm of about 75cm. This gives a theoretical range of about 1.65 metres, but you don't use the tip of a weapon, and you don't fight with your arms fully extended. All things considered, the average soldier armed with a polearm, can probably fight at a distance of about 125cm. Assuming a shoulder height of 1.5 metres for the grizzly bear, therefore arms 1.5 metres long, which can easily deliver enough force to kill you while fully extended, this becomes a problem for the enemy.
So maybe the other guys think this through and decide to use a bow and arrow, after all, it is a big target, problem is, arrows don't pierce steel all that well, and even if it did make it through the armour, it now has to make it through a thick layer of fur, a thick layer of tough bear skin, and a thick layer of fat, before it can do any real damage.
As far as I can see, this only really leaves three alternatives, another bear, an elephant, or fire. Elephants are even more logistically difficult than bears, and not all that common in war, when it comes to other bears, it would likely come down to the bigger, or better armoured bear.
As for fire, there is a reason it wasn't used all that commonly in medieval open warfare, when you have a massive tract of land, packed tight with men, trampled grass, and beasts of war, a fire can be difficult to control, and could just as easily end up killing your own men as it could the enemy.
As for logistics, I have a couple of things I would like to point out.
First off, as "Thomas Pornin" pointed out in an earlier post, grizzly bears eat 75% plant matter, berries, roots and whatnot.
Second of all, even if a bear cost as much to feed as ten men, it could well still be worth it, as they could easily be worth more than ten men on the battlefield, especially when used at the right times.
And third of all, even if you were not willing to feed fallen men of the other side to your bears, which could be for any number of reasons, you could still feed the enemies fallen beats of burden. The horses they ride, and the cattle and donkeys used to drag their supply wagons.
My final point on logistics, is that if you were to send the bears ahead for scouting, some time could be dedicated to allowing the bear to forage, or even hunt, as this would lighten the load on your supplies.
When it comes to tactics, there are several purposes a bear could fulfill.
As scouts, bears might suprise you with how quietly they could make their way through the woods. They would also easily cover ground most people may have trouble with, and would be able to hold their own if they encountered an enemy scout. Obviously you would still need to have at least one person with them, to relay gathered information and whatnot.
On the battleground, while they would be powerful weapons, what would truly make them valuable, would be the fact that your enemy knows they need to be killed, and this will take a large number of soldiers to accomplish. And while twenty enemy men reach your front line concentrated on killing the bear, they are left relatively open to a counter attack by your more traditional forces. Allowing you to use the bear as a sort of anvil, and your men as the hammer.
As a front line unit, they would likely be invaluable to holding the line, as they would be near impossible to push back, even by a cavalry charge, as they are heavier and stronger than even traditional heavy cavalry, and likely wouldn't have much of an issue breaking a horses neck.
They could also serve as a kind of heavy cavalry, accompanied by handlers rather than ridden. The impact they had on the front line of the enemy would be considerable, and quite probibly measured by the tens of enemies fallen per bear.
During any kind of siege, they can take on an additional purpose, properly equipped they could be turned into a kind of living battering ram, generating more force on impact than a team of men.
Additionally, they are, despite their size, capable climbers, and could scale low walls erected by the enemy.
However their is an alternative you could use for your fictional world, which could quite probibly fix any logistics problem you encounter. Giant ground sloths.
These varied in size depending on the species, from the size of a large ape, to the size of a small Elephant, and we're built much like bears.
This solution providers a couple of pros and cons, listed below.
They where herbivores, meaning they where easy to feed, they could simply browse on trees as they went.
They had a network of bones under their considerable hide, which served a perfective purpose, and have been compared to chainmail.
Well normally placid, and thus easily controlled along the journey, unlike horses, they are built for fight, not flight. This means that during battle they would instinctively swipe at the guys pointing the pointy things at them.
However, they still would not have made the best mounts, as, due on part to their large claws, were probably knuckle walkers.
They were also likely considerably slower than a bear of comparable size, they simply weren't built for speed.
As for actually getting to the point where you could take one of these animals into battle, well, a ground sloth likely lived in small groups and so may be easier to tame and domesticate, but as suggested by other posters, you could have the bears in your world be social creatures, it's really up to you.
