How can I set aspect ratio constraints programmatically in iOS?

Like this. Try once.

[self.yourview setTranslatesAutoresizingMaskIntoConstraints:NO];
[self.yourview addConstraint:[NSLayoutConstraint
                                  constraintWithItem:self.yourview
                                  attribute:NSLayoutAttributeHeight
                                  relatedBy:NSLayoutRelationEqual
                                  toItem:self.yourview
                                  attribute:NSLayoutAttributeWidth
                                  multiplier:(self.yourview.frame.size.height / self.yourview.frame.size.width)
                                  constant:0]];

or in the place of (self.yourview.frame.size.height / self.yourview.frame.size.width) you can use any float value. Thanks.

Swift 3.0 -

self.yourview!.translatesAutoresizingMaskIntoConstraints = false
self.yourview!.addConstraint(NSLayoutConstraint(item: self.yourview!,
                                          attribute: NSLayoutAttribute.height,
                                          relatedBy: NSLayoutRelation.equal,
                                          toItem: self.yourview!,
                                          attribute: NSLayoutAttribute.width,
                                          multiplier: self.yourview.frame.size.height / self.yourview.frame.size.width,
                                          constant: 0))

Swift 3:

yourView.addConstraint(NSLayoutConstraint(item: yourView,
                                          attribute: .height,
                                          relatedBy: .equal,
                                          toItem: yourView,
                                          attribute: .width,
                                          multiplier: 9.0 / 16.0,
                                          constant: 0))

Layout Anchors is the most convenient way to set constraints programmatically.

Say you want to set 5:1 aspect ratio for your button then you should use:

button.heightAnchor.constraint(equalTo: button.widthAnchor, multiplier: 1.0/5.0).isActive = true

Here's the full code:

class ViewController: UIViewController {

    override func viewDidLoad() {
        super.viewDidLoad()

        let button = UIButton(type: .custom)
        button.setTitle("Login", for: .normal)
        button.backgroundColor = UIColor.darkGray

        self.view.addSubview(button)

        button.translatesAutoresizingMaskIntoConstraints = false

        let margins = view.layoutMarginsGuide

        button.leadingAnchor.constraint(equalTo: margins.leadingAnchor, constant: 20.0).isActive = true
        button.trailingAnchor.constraint(equalTo: margins.trailingAnchor, constant: -20.0).isActive = true
        button.bottomAnchor.constraint(equalTo: margins.bottomAnchor, constant: -20.0).isActive = true
        button.heightAnchor.constraint(equalTo: button.widthAnchor, multiplier: 1.0/5.0).isActive = true
    }

}

Here're results achieved with code written above. You can see that button keeps its 5:1 aspect ratio across various devices: