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Research Papers

# A Deterministic Lagrangian-Based Global Optimization Approach for Quasiseparable Nonconvex Mixed-Integer Nonlinear Programs

[+] Author and Article Information

Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213

Jeremy J. Michalek

Department of Mechanical Engineering and Department of Engineering and Public Policy, Carnegie Mellon University, Pittsburgh, PA 15213

Factorable programs refer to a class of nonlinear programming problems in which the objective function and constraints are defined in terms of factorable functions. A factorable function is any function that can be formed by taking recursive sums and products of univariate functions.

The convex envelope is tightest possible convex underestimator of a nonconvex function. A convex underestimator of a nonconvex function is a convex function, which lies below the original function.

Since for the general case the objective and constraints are nonconvex, strong duality does not hold, and a duality gap may exist.

The dual problem has two important characteristics: (1) The dual problem is concave regardless of the nature of the primal. (2) If there exists a duality gap, the dual function is nondifferentiable at every dual optimal solution.

The subgradient method is not a descent method; thus the algorithm should keep track of the best point found and report it as the lower bound.

The dual optimal solution is usually primal infeasible but can be employed as a good starting point for solving the primal locally. In the case of a dual feasible solution, it can be used as an upper bound for the overall problem and the node is fathomed.

For example, $S1=[1 0 0; 0 0 1]$ and $y=[y1, y2, y3]T$ indicates that $y1$ and $y3$ are platform components for the first variant, but $y2$ is not.

There are alternative methods for formulating the consistency constraints; our numerical experiments show that the above formulation leads to the tightest lower bounds after a finite number of iterations.

The proposed reformulation can be obtained from authors or www.ddl.me.cmu.edu.

In a recent comprehensive study, Neumaier et al. (32) solved over 1000 test problems from the literature using commercial global solvers and showed that BARON is the fastest and most robust one.

There are various termination options in BARON , which can be controlled by the user. In this paper, we used the relative termination tolerance, which is $εr=100×(UB−LB)/UB%$.

Randomized subgradient method is an extension of the ordinary method developed for separable problems. In each step, only one random subproblem is solved and the multipliers are updated accordingly (20).

The electric motor optimization formulation is a nonconvex NLP; therefore, to find a valid lower bound, it should be solved using a deterministic global optimizer (e.g., BARON ).

CONOPT is a NLP local solver based on the generalized reduced gradient method.

Since this example is a maximization problem, lower (upper) bounding steps are equivalent to upper (lower) bounding stages of Fig. 1.

While the same marketing model and part-worth values were used for this example, the engineering model is modified to be more suitable for global optimization; the detailed formulation can be obtained from authors or www.ddl.me.cmu.edu.

Discrete and continuous optimizer (DICOPT ) is a solver for optimizing convex MINLPs using outer approximation.

J. Mech. Des 131(5), 051009 (Apr 07, 2009) (8 pages) doi:10.1115/1.3087559 History: Received August 25, 2008; Revised January 17, 2009; Published April 07, 2009

## Abstract

We propose a deterministic approach for global optimization of nonconvex quasiseparable problems encountered frequently in engineering systems design. Our branch and bound-based optimization algorithm applies Lagrangian decomposition to (1) generate tight lower bounds by exploiting the structure of the problem and (2) enable parallel computing of subsystems and use of efficient dual methods. We apply the approach to two important product design applications: (1) product family optimization with a fixed-platform configuration and (2) single product design using an integrated marketing-engineering framework. Results show that Lagrangian bounds are much tighter than the factorable programming bounds implemented by the commercial global solver BARON , and the proposed lower bounding scheme shows encouraging robustness and scalability, enabling solution of some highly nonlinear problems that cause difficulty for existing solvers. The deterministic approach also provides lower bounds on the global optimum, eliminating uncertainty of solution quality inherent to popular applications of stochastic and local solvers.

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## Figures

Figure 1

Functional dependence table: (a) arrowhead structure for the original problem, (b) introducing local copies of linking variables (yi) and consistency constraints (c), and (c) relaxing the coupling constraints (g,c) and applying Lagrangian decomposition

Figure 2

Overview of the proposed approach

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