I wanted to try a more naive/slow/sledge-hammer approach of just setting up the cost function and throwing it at an optimizer.

I couldn't get it to converge and I am not sure if it's just the library I am using being buggy (numeric.js) or my cost function is just bad.

Cost:

    const cost = (a, b) => {
      // If a encloses b, returns 0, otherwise a positive cost
      const dx = b.x - a.x;
      const dy = b.y - a.y;
      return Math.max(
        0,
        -a.r + (b.r + Math.sqrt(dx * dx + dy * dy))
      );
    }
    let iter = 0;
    const costAll = ([x, y, r]) => {
      // Cost of enclosing all balls (which should be 0) plus cost of radius (which should get us the smallest ball)
      const a = { x, y, r };
      const radiusCost = Math.abs(a.r);
      let encloseCost = 0;
      for (let i = 0; i < B.length; i++) {
        encloseCost += cost(a, B[i]);
      }  
      const totalCost = encloseCost + radiusCost;
      return totalCost;
    }

Demo:

https://codepen.io/anon/pen/ZyEaaQ?editors=0010

EDIT: demo should work correctly after mbostock's hint below!