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Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 �� b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

Simplified: 40 - 10 - 5

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Title: on existence of solutions to non-convex minimization problems.

Abstract: We provide new sufficient conditions for the finiteness of the optimal value and existence of solutions to a general problem of minimizing a proper closed function over a nonempty closed set. The conditions require an asymptotically bounded decay of a function, a relaxation of p-supercoercivity, and a certain relation for the asymptotic cone of the constraint set and the asymptotic function of the objective function. Our analysis combines these conditions with a regularization technique. We refine the notion of retractive directions of a set, extend its definition to functions, and establish some basic relations for such directions for both sets and functions. Using these tools, we provide existence of solutions results that generalize many of the results in the literature for both non-convex and convex problems.

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