No, you can't just discard to pay the cost. You also have to be able to "choose X" to even attempt to 'trigger' the Action and play the card, as per the FAQ:(1.9) Choosing Targets
The word target is used to indicate that an effect is directing a player to choose 1 or more cards for an effect to resolve on. Not every effect that resolves on a card is targeted. An effect that resolves on 1 or more cards without specifically using the word “choose” is not a targeted effect.
A player cannot trigger a card effect that requires him to choose a character, support card, or story card if there is no card of that type that he is able to choose.
For example, a player could not play Opening the Limbo Gate (Core Set F116) unless every player’s discard pile contained at least one character card.
In addition, a player cannot trigger a card effect that requires him to choose a certain number of targets if there are not enough valid targets available.
Opening the Limbo Gate (Core, F116) reads:Play during your operations phase.
Action: Choose and put into play one character from each player's discard pile.
Note that if Desperate Search's effect had read "Action:
Discard X cards to choose up to
X other cards in your discard pile and add them to your hand..." then I believe you could discard as many cards as you like with an empty discard pile, choosing to return 0 other cards from the discard pile, as per the FAQ:(2.6) The Words “Up To”
Any card effect that involves choosing multiple targets (including choosing multiple cards to draw or discard, or choosing multiple tokens) can only resolve if the exact number of targets, cards, or tokens can be chosen.
For example: The event Byakhee Attack (Core Set F95) reads: “Action:
Each opponent chooses and discards 2 cards from his hand, if able.” If an opponent has exactly 1 card in his hand, he is not affected by Byakhee Attack because he cannot choose 2 cards to discard.
Some cards have effects that include the words “up to,” such as “choose up to 2 characters” or “discard up to 2 cards.” The inclusion of this phrase allows an effect to resolve even if there fewer than the specified number of targets, cards, or tokens available.
For example, the event Dr. Carson’s Treatment (Core Set F39) reads:
Choose up to 2 insane characters. Restore and ready those characters.” The player playing this event can choose 1 or 2 insane characters as targets. If this event did not include the words “up to” the player could not choose only 1 insane character, he could only choose exactly 2 insane characters.
See also this discussion
on the difference between Byakhee Attack (targeted) and Julia Brown, Oddly Amphibious (non-targeted).
Regardless, I believe you could alternatively use Desperate Search to say 'X' is 0, discarding 0 cards to choose 0 other cards, and then just use the second effect to draw 1 card.
Edited by jasonconlon, 19 July 2014 - 07:38 PM.