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Technical Report Subgradient Optimization for Convex Multiparametric Programming

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Technical Report Subgradient Optimization for Convex Multiparametric Programming Technical Report Subgradient Optimization for Convex Multiparametric Programming J.T. Leverenz, H. Lee, M.M. Wiecek Department of Mathematical Sciences Clemson University Clemson, SC [email protected] December 15, 2015 Abstract In this paper we develop a subgradient optimization methodology for convex multiparametric non- linear programs. We define a parametric subgradient, extend some classical optimization results to the multiparametric case, and design a subgradient algorithm that is shown to converge under tradi- tional conditions. We use this algorithm to solve two illustrative example problems and demonstrate its accuracy and efficiency by comparing it to a state-of-the-art multiparametric nonlinear programming algorithm. 1 Introduction The area of multiparametric programming (MPP) has received increased attention in recent years as a means of studying and solving optimization problems in which some of the problem data is unknown or can change. It has recently been applied to problems in model predictive control [16,21,27], multiobjective programming [23], and multilevel hierarchical and decentralized optimization [9]. In these problems the unknown data is represented by parameters contained within a known set and solutions are mappings from the parameter space to optimal decision variable and objective function values. Specifically, MPP involves finding three pieces of information [6]: 1. An exact or approximate expression for the optimal value as a function of the parameters. 2. An exact or approximate expression for the optimal decision variables as functions of the parameters. 3. The partition of the parameter space into critical regions for which the expressions in 1 and 2 are valid. The virtue of parametric solutions to these problems lies in mapping the full range of solutions without knowing the exact conditions represented by the parameters. As conditions change, the optimal solution can quickly be determined by simple function evaluations, bypassing the need for expensive or time-consuming re-optimization of the entire system. This differs from sensitivity analysis, which is focused on posterior evaluation of the local region around a particular, static solution [3, 10]. By contrast, multiparametric optimization methods seek to provide a full map of optimal solutions over the entire parameter space [13,28]. 1 1.1 State of the Art Exact solutions to multiparametric linear (mp-LP) and quadratic problems (mp-QP) can be computed using the Karush-Kuhn-Tucker (KKT) conditions [13, 28]. Methods for general nonlinear problems (mp- NLP) produce approximate solutions and can be broadly categorized into three areas: pathfollowing or homotopy methods [11,15], parameter space partition methods [2,25], and problem approximation methods [7, 8, 20]. A survey of recent advances in the latter two areas can be found in [6]. Homotopy methods are only used in the single parameter case. They determine a continuous \path" of KKT points created as the parameter moves along an interval (i.e., the parameter space). Partition methods solve mp-NLPs by dividing the parameter space into smaller sets to approximate the critical regions of the optimal solution. The problem is then solved as a standard NLP at the vertices of each set and optimal value and decision functions are interpolated from the results. Approximation methods replace the mp-NLP with a series of mp-LP or mp-QP problems that can then be solved for exact solutions. Additional approximations can be made until the desired accuracy is achieved. These methods all proceed by solving the primal problem: dual problems are ignored except in some cases to provide parametric bounds on solutions [2,10]. Dual problems play an important role in optimiza- tion and exploring their use with parameters can open new possibilities for solution strategies to mp-NLPs. In particular, subgradient methods are often paired with dual problems. These algorithms are a popular choice to solve nonlinear programs: they tend to be straightforward to implement and are supported by a deep background of theory while remaining a rich field for ongoing research. Subgradient optimization was originally developed for nondifferentiable functions [1, 29] and has been applied in a variety of fields, including solving complex systems [14, 22], nonconvex NLPs [4, 12], and mixed-integer problems [29]. Frechet subgradients of mp-NLPs have previously been explored in [24], but no work has been done on a characterization of the subgradient that can be utilized in an algorithm for solving mp-NLPs. The goal of this paper is to develop subgradient optimization for convex mp-NLPs. To do this we define a parametric subgradient and extend some classical optimization results to the multiparametric case, design an algorithm, and demonstrate that this algorithm can produce results competitive with those of a state-of-the-art mp-NLP approximation method. The outline of the paper is as follows. In Section 2 we state the formulation of an mp-NLP and prove some theoretical results. Section 3 introduces a parametric subgradient algorithm and provides conditions for its convergence. Section 4 demonstrates the algorithm on two example problems. Results are summarized in Section 5. 2 Problem Formulation and Theory This section contains the results needed to define a parametric subgradient and to use it to identify an optimal solution of an mp-NLP. Many current results related to subgradient optimization are extended to the multiparametric case. The formulation of the general mp-NLP depends on whether decision variables are treated as implicit or explicit functions of the parameters. The implicit formulation is commonly found in the literature (e.g. [6, 10, 15]) and is used for applications. The explicit formulation is less commonly used, but is useful for proving many of the theoretical results in this paper. Both formulations are presented in Section 2.1. The parametric subgradient is defined in Section 2.2 and several results are extended to the multiparametric case. The multiparametric Lagrangian dual problem is introduced in Section 2.3 with weak and strong duality relationships shown to hold in the presence of parameters. 2 2.1 Formulation The implicit formulation of an mp-NLP is given by: f ∗(t) = min f(x; t) x st g(x; t) ≤ 0 h(x; t) = 0 (2.1) x 2 S t 2 Ω n p n p where x 2 S ⊂ R , t is a vector of parameters, the set Ω ⊂ R is the parameter space, f : R × R ! R, n p m n p l g : R × R ! R , and h : R × R ! R . For t 2 Ω the feasible set is determined by X(t) = fx 2 S : g(x; t) ≤ 0; h(x; t) = 0g. We assume that X(t) 6= ; 8t 2 Ω, that Ω is a closed, connected set, and that all functions and sets are continuous and convex. The definition of connected set is given by Fiacco [10] in which Ω is connected if there do not exist disjoint open sets A1 and A2 such that Ω ⊂ A1 [ A2,Ω \ A1 6= ;, and Ω \ A2 6= ;. This does not present an obstacle to the practical solution of (2.1) since a disconnected set can be regarded as the union of disjoint sets Ai, each of which can be treated as a separate parameter space. Likewise, the condition that X(t) 6= ; 8t 2 Ω is based on the assumption that infeasible parameter values t can be removed from the parameter space, leaving a collection of disjoint feasible parameter spaces. This is a strong assumption from a practical standpoint since it can not always be known in advance that X(t) 6= ; for the chosen Ω, but the identification and removal of infeasible parameter values is a separate issue beyond the scope of this paper. The solution to (2.1) maps each element of Ω to an optimal solution x∗ and optimal value f ∗. The result of this mapping is an optimal decision function x∗(t) and an optimal value function f ∗(t) = f (x∗(t)). These solutions are typically piecewise functions of the parameters and they partition Ω into critical regions defined by subsets of Ω associated with each part of x∗(t) and f ∗(t). Problem (2.1) is the most convenient and common way to formulate an mp-NLP. However, for the purposes of proving the theoretical results in this paper it is more advantageous to reformulate (2.1) so that the variables x are explicit functions of the parameters t. In the explicit formulation of the general mp-NLP, the variables x are treated as functions x(t) belonging to a vector-valued function space. The following theorem [10, Theorem 2.2.2] motivates the choice of function space. n Theorem 2.1. Consider (2.1) where X(t) is a continuous point-to-set mapping from Ω to R and f is a n ∗ continuous function on R × Ω. Then the optimal value function f (t) is continuous on Ω and the optimal decision functions x∗(t) are upper semicontinuous. With this result an appropriate vector-valued function space equipped with an inner product can be chosen and utilized for the explicit formulation of (2.1). 2 n Definition 2.2. The Hilbert space L (Ω; R ) is the space of square-integrable, vector-valued functions on 2 n n 2 1 2 Ω. For x(t) 2 L (Ω; R ), x :Ω ! R . When n = 1, L (Ω; R ) is written as L (Ω). The inner product is Z T 2 n hx1(t); x2(t)i = x1(t) x2(t)dt for x1(t); x2(t) 2 L (Ω; R ) Ω and the induced norm is Z kx(t)k2 = x(t)T x(t)dt Ω 2 For f1(t); f2(t) 2 L (Ω) we say that f1(t) ≤ f2(t) if the inequality holds for every t 2 Ω.
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