# Second-Order Cone Relaxations for Binary Quadratic Polynomial Programs.

### Full-text

Juan C Vera, Jan 21, 2014 Available from:-
- "Problem (7) is one such example that leads to a somewhat concise semi-definite program. We refer the interested reader to Lasserre (2002) and Ghaddar et al. (2011) for a hierarchy of polynomial size semi-definite programming relaxation of mixed-integer quadratic programs for which the integrality gap is known to converge to 1. Based on Proposition 2, it is therefore theoretically possible to find a semi-definite programming model of polynomial size that will generate a solution within a constant factor of the optimal one. Unfortunately, this might often be of little practical relevance. "

##### Article: Robust Optimization of Sums of Piecewise Linear Functions with Application to Inventory Problems

[Show abstract] [Hide abstract]

**ABSTRACT:**Robust optimization is a methodology that has gained a lot of attention in the recent years. This is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models. Unfortunately, the second property is usually lost when the cost function that needs to be “robustified” is not concave (or linear) with respect to the perturbing parameters. In this paper, we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set. Given that these problems are known to be intractable, we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed-integer linear program and relate this scheme to methods that are based on exploiting affine decision rules. Our new scheme gives rise to two tractable models that respectively take the shape of a linear program and a semi-definite program, with the latter having the potential to provide solutions of better quality than the former at the price of heavier computations. We present conditions under which our approximation models are exact. In particular, we are able to propose the first exact reformulations for a robust (and distributionally robust) multi-item newsvendor problem with budgeted uncertainty set and a reformulation for robust multi-period inventory problems that is exact whether the uncertainty region reduces to a L1-norm ball or to a box. An extensive set of empirical results will illustrate the quality of the approximate solutions that are obtained using these two models on randomly generated instances of the latter problem. -
- "Linear refers to generating linear inequalities that are added to the master problem by using a non-negative multiplier. SOCP refers to generating linear inequalities that are added to the master problem by using a polynomial multiplier that is in P 1 (B) [10]. Quadratic refers to generating quadratic inequalities similar to the previous examples described. "

##### Conference Paper: An Iterative Scheme for Valid Polynomial Inequality Generation in Binary Polynomial Programming.

[Show abstract] [Hide abstract]

**ABSTRACT:**Semidefinite programming has been used successfully to build hierarchies of convex relaxations to approximate polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small sizes. We propose an iterative scheme that improves the semidefinite relaxations without incurring exponential growth in their size. The key ingredient is a dynamic scheme for generating valid polynomial inequalities for general polynomial programs. These valid inequalities are then used to construct better approximations of the original problem. As a result, the proposed scheme is in principle scalable to large general combinatorial optimization problems. For binary polynomial programs, we prove that the proposed scheme converges to the global optimal solution for interesting cases of the initial approximation of the problem. We also present examples illustrating the computational behaviour of the scheme and compare it to other methods in the literature.Integer Programming and Combinatoral Optimization - 15th International Conference, IPCO 2011, New York, NY, USA, June 15-17, 2011. Proceedings; 01/2011 - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper presents a canonical dual method for solving a quadratic discrete value selection problem subjected to inequality constraints. The problem is first transformed into a problem with quadratic objective and 0-1 integer variables. The dual problem of the 0-1 programming problem is thus constructed by using the canonical duality theory. Under appropriate conditions, this dual problem is a maximization problem of a concave function over a convex continuous space. Numerical simulation studies, including some large scale problems, are carried out so as to demonstrate the effectiveness and efficiency of the method proposed.