Of course it is.
He is specifically and explicitly made Commander In Chief of America's military forces.
Not "Commander In Part" or "Commander To A Certain Extent."
The law requires him to update Congress on his decisions and discuss his decisions with them (which he should obviously be doing anyway) and requires him to rely on Congress to fund the military budget, but as Commander In Chief he has full discretion to send our military forces anywhere in the world for any purpose that national security requires according to his lights.
Unless the word "Commander" has some secret meaning known only to Paulestinians to which lesser beings are not privy.
Actually, it seems that the Paul supporters/deaconblues have it right. The legislature has the "What" power, while the Executive branch handles the "How" implementation. Congress declares War, and President implements it.
The Commander of a battalion has no power to just go running off with it to do whatever he wants, and neither does the POTUS have the power to do whatever he wants with the entire military.
My points of contention were with your earlier statements that the President "is entitled to deploy troops anywhere outside the United States he desires for any reason" and that he has "complete authority".
Hamilton stated in the Federalist No. 69: "The President is to be commander-in-chief of the army and navy of the United States. In this respect his authority would be nominally the same with that of the king of Great Britain, but in substance much inferior to it. It would amount to nothing more than the supreme command and direction of the military and naval forces, as first General and admiral of the Confederacy; while that of the British king extends to the DECLARING of war and to the RAISING and REGULATING of fleets and armies, all which, by the Constitution under consideration, would appertain to the legislature."
It is my interpretation that the framers fully intended the President to have full control over the Command of the military, but specifically denied him the power to commit troops to war without a Congressional declaration. Further, it is my understanding that the War Powers Act of 1973, (the Constitutionality of which is still open for debate), clarified Presidential responsibilities when he chooses to place troops in harm's way, without a formal declaration of war.
Regards.