This is a survey on the computational complexity of nonlinear mixedinteger optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained fully polynomial time approximation schemes in fixed dimension, and to strongly polynomialtime algorithms for special cases.

On the Complexity of Nonlinear Mixed-Integer Optimization

Increasing population, economic development, and environmental changes imply that maintaining the water supply-demand balance will remain a top priority. Water resource systems may need to be expanded in order to respond to demand growth. Capacity expansion studies can be used to answer the question of what the optimal expansion size, timing and location of new infrastructure should be. This thesis develops and applies capacity expansion optimisation modelling approaches. We begin with the 'Economics of Balancing Supply and Demand’ (EBSD) planning framework used by the water industry since 2002 in England. The base model is formulated as a mixed integer linear programming optimisation model that selects the least cost annual schedule of supply and demand management options that meet forecasted demand over the planning horizon. Custom water saving profiles are allowed for demand management options. Multiple demand scenarios are considered to ensure the supply-demand balance is preserved under different demand conditions and that operating costs of selected options are accurately assessed. The base deterministic EBSD model is applied to the water companies of South East England (the WRSE area). Various extensions to the EBSD framework are then proposed and implemented. The model formulation is first expanded to incorporate a generic cost estimate for options not yet proposed in water company resources management plans. This allows to extend the WRSE network with new inter-company transfers for which costs are represented by a concave cost curve approximated by a piece-wise linear function. Considering additional interconnections allows evaluating the financial implications of further interconnectivity in the WRSE area. Next, an extension is proposed to improve the application of the stochastic version of the EBSD approach. The proposed method allows to identify the set of future capacity expansions that withstand uncertainty of supply and demand estimates and still achieve a required reliability. The method consists of an iterative process: at each iteration the EBSD optimisation model is run and the reliability of the solution set (supply-demand schedules) is tested under Monte Carlo simulation. Ad-hoc model constraints are introduced at each iteration to enable the EBSD model to exclude unreliable solutions identified at previous iterations. Next, the English price-cap regulatory process is represented within a modified EBSD model formulation. The model identifies future capacity expansions that maximise water company profit under constraints on the maximum price that can be charged to customers and on the allowed rate of return. The incentive schemes that the regulator uses to reward (or penalise) companies for out- (or under-) performance, are also represented. The goal is to help explain how the current regulatory system of incentives motivates water company investment decisions. Two further extensions are then presented in the appendixes. The first one allows the EBSD model formulation to be extended so that costs of activated schemes are accounted over the schemes’ useful life, beyond the typical 25-30 year planning horizon. This eliminates biased comparisons of schemes with different economic lifetimes. With the second extension, a diverse set of supply-demand schedules are generated, that solve the capacity expansion problem and are sufficiently ‘close’ (in terms of costs) to the least-cost solution. Generating multiple near-optimal solutions gives an idea of what alternative plans are available in addition to the leastcost one. This allows the consideration other un-modelled factors or strategic priorities in the decision making process.

Capacity expansion modelling to aid water supply investment decisions

In this paper we are concerned with the problem of boundedness and the existence of optimal solutions to the constrained integer optimization problem. We present necessary and sufficient conditions for boundedness of either a faithfully convex or quasi-convex polynomial function over the feasible set contained in $${\mathbb {Z}^n}$$
 , and defined by a system of faithfully convex inequality constraints and/or quasi-convex polynomial inequalities. The conditions for boundedness are provided in the form of an implementable algorithm, terminating after a finite number of iterations, showing that for the considered class of functions, the integer programming problem with nonempty feasible region is unbounded if and only if the associated continuous optimization problem is unbounded. We also prove that for a broad class of objective functions (which in particular includes polynomials with integer coefficients), an optimal solution set of the constrained integer problem is nonempty over any subset of $${\mathbb {Z}^n}$$
 .

On boundedness of (quasi-)convex integer optimization problems

In this paper we investigate certain aspects of infeasibility in convex integer programs, where the constraint functions are defined either as a composition of a convex increasing function with a convex integer valued function of n variables or the sum of similar functions. In particular we are concerned with the problem of an upper bound for the minimal cardinality of the irreducible infeasible subset of constraints defining the model. We prove that for the considered class of functions, every infeasible system of inequality constraints in the convex integer program contains an inconsistent subsystem of cardinality not greater than 2n, this way generalizing the well known theorem of Scarf and Bell for linear systems. The latter result allows us to demonstrate that if the considered convex integer problem is bounded below, then there exists a subset of at most 2n−1 constraints in the system, such that the minimum of the objective function subject to the inequalities in the reduced subsystem, equals to the minimum of the objective function over the entire system of constraints.

Minimal infeasible constraint sets in convex integer programs

In this paper we are concerned with the problem of unboundedness and existence of an optimal solution in reverse convex and concave integer optimization problems. In particular, we present necessary and sufficient conditions for existence of an upper bound for a convex objective function defined over the feasible region contained in ${\mathbb{Z}^n}$. The conditions for boundedness are provided in a form of an implementable algorithm, showing that for the considered class of functions, the integer programming problem is unbounded if and only if the associated continuous problem is unbounded. We also address the problem of boundedness in the global optimization problem of maximizing a convex function over a set of integers contained in a convex and unbounded region. It is shown in the paper that in both types of integer programming problems, the objective function is either unbounded from above, or it attains its maximum at a feasible integer point.

Unboundedness in reverse convex and concave integer programming

In this paper, we are concerned with the problem of boundedness in the constrained global maximization of a convex function. In particular, we present necessary and sufficient conditions for boundedness of a feasible region defined by reverse convex constraints and we establish sufficient and necessary conditions for existence of an upper bound for a convex objective function defined over the system of concave inequality constraints. We also address the problem of boundedness in the global maximization problem when a feasible region is convex and unbounded.

Wiesława T. Obuchowska

Papers

On boundedness of (quasi-)convex integer optimization problems

Conditions for boundedness in concave programming under reverse convex and convex constraints

Minimal infeasible constraint sets in convex integer programs

Unboundedness in reverse convex and concave integer programming

Feasibility in reverse convex mixed-integer programming