Can you elaborate on what a "Byronic view of existence" is?

The malevolent universe premise comes in may variations, from the passive cynicism of the used-record salesmen in "High Fidelity" to the sturm und drang of a Wagnerian opera. And while Wile E. Coyote is indeed a comic manifestation of a malevolent universe premise, he is not specifically representative of the Byronic sense of life. He is way too small for that. The Byronic sense of life is grand, not small.

The "Byron" of "Byronic" is George Gordon Byron, one of the second-generation Romantic poets in England. His period of greatest fame lasted from about 1819 until his untimely death in 1824, at the age of 36.

Byron's poems often depict the so-called "Byronic hero," a brilliant, moral, courageous, but misunderstood man forced into the life of an outlaw because of some (usually unspecified) falling-out with society. Without providing details, the poet clearly implies that the hero has been persecuted for what were in fact noble actions.

Here's how that heroic type is described in "The Corsair," my personal favorite of Byron's poems. The hero of this poem is a Mediterranean pirate who makes his living by raiding the shipping from "bad guys," presumably Turkish rulers with dictatorial power and harems of powerless women. Conrad, the hero, is described as:

That man of loneliness and mystery

Scarce seen to smile, and seldom heard to sigh;

Whose name appals the fiercest of his crew,

... Still sways their souls with that commanding art

... The power of Thought—the magic of the Mind!

Linked with success, assumed and kept with skill,

...

Accuse not—hate not—him who wears the spoils.

The hero is not just smart, he's genius; he's not just a leader, he's a commander; he's not just ethical, he is noble. He is heroic; and yet, precisely because of his heroic qualities, he is an outcast from society.

Herein lies the problem, from the standpoint of the Objectivist esthetics.

The point of Byron's portrayal is to suggest that the heroic is the doomed. In Ayn Rand's eloquent summary, the essence of Byron's sense of life is "the belief that man must lead a heroic life and fight for his values even though he is doomed to defeat by a malevolent fate over which he has no control." It is a beautiful, yet exquisitely painful view of existence: it exalts the great, but consigns them metaphysically to a pit of horrors.

Ayn Rand's view is that the moral is the practical: that virtuous action is, metaphysically, the path to success. If you want to succeed, objectively — not in terms of satisfying subjective and self-destructive whims, but in terms of actually flourishing — ethics is the fundamental means to do so.

It is, of course, possible for a virtuous man — a hero, even — to fail through no fault of his own. However, in art and literature one portrays that which one considers to be the metaphysical essence of man. Hence, to portray the hero as routinely doomed is to portray the hero as metaphysically doomed.

Byron's poetry is in my view brilliant. It beautifully concretizes a tragic, yet heroic sense of life. Unfortunately, although I admire his poetry greatly, his sense of life is more Nietzsche than Ayn Rand.

The original questioner is mistaken to suggest that the Byronic hero is:

... a person who believes in a cause or works hard for a goal but never reaches it?

That description fits half the population! Almost everyone "believes in a cause or works hard for a goal but never reaches it." You can believe in a cause, see it fail, yet not be malevolent-universe. The point of the Byronic sense of life is the view of a heroic man inherently at war with the world because of his virtues.

In the comments, the questioner asks:

If John Galt had not attempted to "stop the motor of the world," would he be a Byronic Hero?

This question seems to confuse ethics with metaphysics, i.e., ethical values with metaphysical values. The Lexicon topic, "Byronic View of Existence," is concerned with metaphysical values. Refer to the Lexicon topic of "Metaphysical Value-Judgments" for further discussion of metaphysical values. In esthetics, Objectivism advocates works of art that concretize metaphysical values, as providing the greatest artistic significance and value to man. The issue is not whether or not a character is Byronic, but whether or not the universe in which that character functions is Byronic:

There are Romanticists whose basic premise, in effect, is that man possesses volition in regard to consciousness, but not to existence, i.e., in regard to his own character and choice of values, but not in regard to the possibility of achieving his goals in the physical world.

Applying this to the setting in Atlas Shrugged, I would say that it would be a Byronic view of existence if Ayn Rand had depicted John Galt as heroically striving to stop the motor of the world, but failing dismally. Perhaps at the end of the story, the world simply goes on without him, as if his existence and his greatness of character made not one iota of difference to anyone else -- not because he didn't try, but because it allegedly doesn't matter in the "grand scheme of things."

The question of the difference between John Galt and the Byronic sense of life is a very interesting one. Ayn Rand did write one character who represented the idea that the good, as such, is metaphysically doomed: Dominique Francon. Dominique's sense of life is essentially Byronic.

Although in some sense John Galt is an "outcast" in self-imposed exile throughout most of the book, neither Galt himself, nor the author, treats Galt as inherently doomed. Indeed, he is highly successful in achieving his goal, to stop the motor of the world. Although Galt withdraws from the world, he does not give up on the world as a Byronic hero would. He does not seethe with hatred for the looters, he merely prevents them from looting him any more (nor from the others who join his strike). He does not inflict harm on those who remain in the world, he merely withdraws his sanction and support. In short, he does not invest the evil with metaphysical significance, the way a Byronic hero would.

Galt's self-exile from the world is presented as an unnatural condition, the result of the irrational philosophy of his age, not inherent in human nature. And of course the novel only ends after the evil has collapsed, Galt and Dagny are together, and he announces that it is time to go back and reclaim their proper position in the world.