Undecidability and hardness in mixed-integer nonlinear programming

We survey two aspects of mixed-integer nonlinear programming which have attracted less attention (so far) than solution methods, solvers and applications: namely, whether the class of these problems can be solved algorithmically, and, for the subclasses which can, whether they are hard to solve. We start by reviewing the problem of representing a solution, which is linked to the correct abstract computational model to consider. We then cast some traditional logic results in the light of mixed-integer nonlinear programming, and come to the conclusion that it is not a solvable class: instead, its formal sentences belong to two different theories, one of which is decidable while the other is not. Lastly, we give a tutorial on computational complexity and survey some interesting hardness results in nonconvex quadratic and nonlinear programming.

On a solvable class of product-type systems of difference equations

It is shown that the following class of systems of difference equations zn+1 = ?zanwbn, wn+1 = ?wcnzdn-2, n ? N0, where a,b,c,d ? Z, ?, ?, z-2, z-1, z0,w0 ? C \ {0}, is solvable, continuing our investigation of classification of solvable product-type systems with two dependent variables. We present closed form formulas for solutions to the systems in all the cases. In the main case, when bd ? 0, a detailed investigation of the form of the solutions is presented in terms of the zeros of an associated polynomial whose coefficients depend on some of the parameters of the system.

Monotone Temporal Planning: Tractability, Extensions and Applications

This paper describes a polynomially-solvable class of temporal planning problems. Polynomiality follows from two assumptions. Firstly, by supposing that each sub-goal fluent can be established by at most one action, we can quickly determine which actions are necessary in any plan. Secondly, the monotonicity of sub-goal fluents allows us to express planning as an instance of STP≠ (Simple Temporal Problem with difference constraints). This class includes temporally-expressive problems requiring the concurrent execution of actions, with potential applications in the chemical, pharmaceutical and construction industries. We also show that any (temporal) planning problem has a monotone relaxation which can lead to the polynomial-time detection of its unsolvability in certain cases. Indeed we show that our relaxation is orthogonal to relaxations based on the ignore-deletes approach used in classical planning since it preserves deletes and can also exploit temporal information.