By Elizabeth Karas, Ademir Ribeiro, Claudia Sagastizabal

For fixing nonsmooth convex restricted optimization difficulties, we advise an set of rules which mixes the guidelines of the proximal package deal tools with the clear out approach for comparing candidate issues. The ensuing set of rules inherits a few beautiful beneficial properties from either methods. at the one hand, it permits potent keep watch over of the dimensions of quadratic programming subproblems through the compression and aggregation strategies of proximal package deal tools. nevertheless, the clear out criterion for accepting a candidate element because the new iterate is usually more uncomplicated to meet than the standard descent situation in package deal equipment. a few encouraging initial computational effects also are stated.

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**Example text**

Deﬁne the linear operator T by 1 (T h)(x) = h(y)K(x, y) dy. 0 Let r be the minimum value of f /g on [0, 1]. If f − rg is not identically zero, then T 2 (f − rg) is positive on [0, 1], contradicting T 2 (f − rg) = f − rg. B5. Find a polynomial identity with integer coeﬃcients satisﬁed by the six distances. Obtain a contradiction modulo a small power of 2. B6. It suﬃces to show that (a, b, c) with 0 < a ≤ b ≤ c can be transformed into (b , r, c ) where b = qa + r and 0 ≤ r < a. This is accomplished by a sequence of transformations dictated by the binary expansion of q.

39 Hints: The Fiftieth Competition (1989) The Fiftieth William Lowell Putnam Mathematical Competition December 2, 1989 A1. Note that 100k − 1 = (10k + 1)(10k − 1). A2. Divide the rectangle into two parts by the diagonal line ay = bx. A3. Show that the fractional linear transformation 1 } and { z : |z| > 1 } ∪ {∞}. 11−10iz 11z+10i interchanges { z : |z| < A4. Let coin ﬂips determine digits past the decimal point in the binary expansion of a real number. A5. An interval of what length is needed to contain all the distances |p − v|?

For a given positive integer m, ﬁnd all triples (n, x, y) of positive integers, with n relatively prime to m, which satisfy (x2 + y 2 )m = (xy)n . (page 154) A4. Let f be an inﬁnitely diﬀerentiable real-valued function deﬁned on the real numbers. If 1 n2 , n = 1, 2, 3, . . , f = 2 n n +1 compute the values of the derivatives f (k) (0), k = 1, 2, 3, . . (page 155) A5. For each positive integer n, let an = 0 if the number of 1’s in the binary representation of n is even, 1 if the number of 1’s in the binary representation of n is odd.

### A bundle-filter method for nonsmooth convex constrained optimization by Elizabeth Karas, Ademir Ribeiro, Claudia Sagastizabal

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