In this paper we investigate the computational complexity of solving ordinary differential equations (ODEs) y ' = p ( y ) over unbounded time domains, where p is a vector of polynomials. Contrarily to the bounded (compact) time case, this problem has not been well-studied, apparently due to the "intuition" that it can always be reduced to the bounded case by using rescaling techniques. However, as we show in this paper, rescaling techniques do not seem to provide meaningful insights on the complexity of this problem, since the use of such techniques introduces a dependence on parameters which are hard to compute.We present algorithms which numerically solve these ODEs over unbounded time domains. These algorithms have guaranteed accuracy, i.e. given some arbitrarily large time t and error bound e as input, they will output a value y ? which satisfies ? y ( t ) - y ? ? ? e . We analyze the complexity of these algorithms and show that they compute y ? in time polynomial in several quantities including the time t, the accuracy of the output e and the length of the curve y from 0 to t, assuming it exists until time t. We consider both algebraic complexity and bit complexity.
The reason to focus on the Lipschitz condition for @math seems to be fundamental when studying the computational complexity of . Indeed it is well-known (see e.g. [Theorem 7.3] Ko91 ) that if @math is not Lipschitz, then the solution of can have arbitrarily high complexity, even if @math is assumed to be polynomial-time (precision-)computable and has a unique solution. The Lipschitz condition plays an instrumental role in the complexity because it is used to derive the number of steps of the algorithm, for example in Picard-Lindel " o f theorem it is used to bound the number of iterations. It was recently shown @cite_10 , following an open problem from Ko [Section 7.2] Ko91 , that if @math is Lipschitz and polynomial-time (precision-)computable, then the solution of over @math can still be PSPACE-complete (but it was already known not to be of higher complexity @cite_8 ).
@cite_10 @cite_8
2102100313 1545819195
We provide sharp worst-case evaluation complexity bounds for nonconvex minimization problems with general inexpensive constraints, i.e. problems where the cost of evaluating enforcing of the (possibly nonconvex or even disconnected) constraints, if any, is negligible compared to that of evaluating the objective function. These bounds unify, extend or improve all known upper and lower complexity bounds for unconstrained and convexly-constrained problems. It is shown that, given an accuracy level @math , a degree of highest available Lipschitz continuous derivatives @math and a desired optimality order @math between one and @math , a conceptual regularization algorithm requires no more than @math evaluations of the objective function and its derivatives to compute a suitably approximate @math -th order minimizer. With an appropriate choice of the regularization, a similar result also holds if the @math -th derivative is merely H "older rather than Lipschitz continuous. We provide an example that shows that the above complexity bound is sharp for unconstrained and a wide class of constrained problems, we also give reasons for the optimality of regularization methods from a worst-case complexity point of view, within a large class of algorithms that use the same derivative information. In answer to Ko's question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomial-space computation tableaux with equally restricted feedback, and show that they are still polynomial-space complete. The same technique also settles Ko's two later questions on Volterra integral equations. We introduce the general and powerful scheme of predicting information re-use in optimization algorithms. This allows us to devise a computationally efficient algorithm for bandit convex optimization with new state-of-the-art guarantees for both Lipschitz loss functions and loss functions with Lipschitz gradients. This is the first algorithm admitting both a polynomial time complexity and a regret that is polynomial in the dimension of the action space that improves upon the original regret bound for Lipschitz loss functions, achieving a regret of @math . Our algorithm further improves upon the best existing polynomial-in-dimension bound (both computationally and in terms of regret) for loss functions with Lipschitz gradients, achieving a regret of @math . In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in @math , @math , and @math biased games, and then use these algorithms to provide strongly polynomial algorithms that find @math , @math , and @math approximations for these norms, respectively.