There is a difference between a hero and a protagonist. There is a difference between sympathy and empathy. There is a difference between relating to a character and connecting with them. From common thieves, mob bosses, assassins, serial killers and myriad monsters and demons, literature is rife with antiheroes and villainous main characters.
The reader need not approve of the character. The character need not have any redeemable qualities whatsoever. You will need to find other tactics to grab your reader's attention.
Morbid curiosity/Macabre - This is partly why the Sharon Tate murders are still talked about more than fifty years later.
Psychological fascination - Why Hannibal Lecter is so iconic. The character is thoroughly deplorable, and yet still the anchor of the series.
Plot/Arc - Though individual characters may have had some redeeming qualities, The Lannisters were deliciously despicable, as were the Corleones and Tattaglias. Fans loved watching the characters lie, swindle and kill each other.
Negative Illustration - You can make a statement about good (or broader social statement) through depicting evil, viz Alex DeLarge. (Admittedly, this narrative makes use of pretty much all the aforementioned devices, and is a master class of dystopian/antihero theme, but for sake of just a character study is befitting.)
Character Concept - Here again, using any/all the above techniques, you devise a character with an interesting concept that titillates your readers, makes them want to explore what makes the character tick, or indulge in their darkness from the safety of the fourth wall, and this list is nearly endless: Count Dracula, Travis Bickle, Char Aznable, Tyler Durden, ad nauseum (quite literally).
So no, the reader doesn't have to approve of the main character at all. There is a reason Jack the Ripper and Mike Meyers capture fans. There is a difference between an ethically good character and a conceptually good character.
And never underestimate the importance of just good writing.
Btw, the tragedy element is pretty basic IMO: it is what could-have-been. Flashes or hints of virtue appear from beneath the mire, never to be realized or allowed to shine. A character who could be likable, or lovable, or admirable, but through flaw (character-driven) or circumstance (plot-driven), never attains redemption, or love or what-have-you.
I would recommend reading up on some psychology; I personally don't believe in such a thing as pure evil or pure good; we are all faceted. So understanding your character's thinking and motivations (not to mention the psychology/desires of your audience) may help you delve into the mind and nature of your character, and breathe a fuller life into your concept.
Then all you gotta do is write the devil out of it. 3:)