Hanif D. Sherali

A hierarchy of relaxation between the continuous and convex hull representations

In this paper a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero-one polynomial programming problem, and then relinearizes it into an extended linear program. It is shown that the strength of the resulting reformulation depends on the degree of the terms used to produce the polynomial program at the intermediate step of this method. In fact, as this degree varies from one up to the number of variables in the problem, a hierarchy of sharper representations is obtained with the final relaxation representing the convex hull of feasible solutions. The reformulation technique readily extends to produce a similar hierarchy of linear relaxations for zero-one polynomial programming problems. A characterization of the convex hull in the original variable space is also available through a projection process. The structure of this convex hull characterization (or its other relaxations) can be exploited to generate strong or facetial valid inequalities through appropriate surrogates in a computational framework. The surrogation process can also be used to study various classes of facets for different combinatorial optimization problems. Some examples are given to illustrate this point. 1. Introduction. This paper describes a technique for generating a hierarchy of polyhedral representations for linear and polynomial zero-one programming problems.In this paper a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero-one polynomial programming problem, and then relinearizes it into an extended linear program. It is shown that the strength of the resulting reformulation depends on the degree of the terms used to produce the polynomial program at the intermediate step of this method. In fact, as this degree varies from one up to the number of variables in the problem, a hierarchy of sharper representations is obtained with the final relaxation representing the convex hull of feasible solutions. The reformulation technique readily extends to produce a similar hierarchy of linear relaxations for zero-one polynomial programming problems. A characterization of the convex hull in the original variable space is also available through a projection process. The structure of this convex hull characterization (or its other relaxations) can be exploited to generate strong or facetial valid inequaliti...

On energy provisioning and relay node placement for wireless sensor networks

Wireless sensor networks that operate on batteries have limited network lifetime. There have been extensive recent research efforts on how to design protocols and algorithms to prolong network lifetime. However, due to energy constraint, even under the most efficient protocols and algorithms, the network lifetime may still be unable to meet the missions requirements. In this paper, we consider the energy provisioning (EP) problem for a two-tiered wireless sensor network. In addition to provisioning additional energy on the existing nodes, we also consider deploying relay nodes (RNs) into the network to mitigate network geometric deficiencies and prolong network lifetime. We formulate the joint problem of EP and RN placement (EP-RNP) into a mixed-integer nonlinear programming (MINLP) problem. Since an MINLP problem is NP-hard in general, and even state-of-the-art software and techniques are unable to offer satisfactory solutions, we develop a heuristic algorithm, called Smart Pairing and INtelligent Disc Search (SPINDS), to address this problem. We show a number of novel algorithmic design techniques in the design of SPINDS that effectively transform a complex MINLP problem into a linear programming (LP) problem without losing critical points in its search space. Through numerical results, we show that SPINDS offers a very attractive solution and some important insights to the EP-RNP problem.

A new reformulation-linearization technique for bilinear programming problems

This paper is concerned with the development of an algorithm for general bilinear programming problems. Such problems find numerous applications in economics and game theory, location theory, nonlinear multi-commodity network flows, dynamic assignment and production, and various risk management problems. The proposed approach develops a new Reformulation-Linearization Technique (RLT) for this problem, and imbeds it within a provably convergent branch-and-bound algorithm. The method first reformulates the problem by constructing a set of nonnegative variable factors using the problem constraints, and suitably multiplies combinations of these factors with the original problem constraints to generate additional valid nonlinear constraints. The resulting nonlinear program is subsequently linearized by defining a new set of variables, one for each nonlinear term. This “RLT” process yields a linear programming problem whose optimal value provides a tight lower bound on the optimal value to the bilinear programming problem. Various implementation schemes and constraint generation procedures are investigated for the purpose of further tightening the resulting linearization. The lower bound thus produced theoretically dominates, and practically is far tighter, than that obtained by using convex envelopes over hyper-rectangles. In fact, for some special cases, this process is shown to yield an exact linear programming representation. For the associated branch-and-bound algorithm, various admissible branching schemes are discussed, including one in which branching is performed by partitioning the intervals for only one set of variables x or y, whichever are fewer in number. Computational experience is provided to demonstrate the viability of the algorithm. For a large number of test problems from the literature, the initial bounding linear program itself solves the underlying bilinear programming problem.

A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique

This paper is concerned with the development of an algorithm to solve continuous polynomial programming problems for which the objective function and the constraints are specified polynomials. A linear programming relaxation is derived for the problem based on a Reformulation Linearization Technique (RLT), which generates nonlinear (polynomial) implied constraints to be included in the original problem, and subsequently linearizes the resulting problem by defining new variables, one for each distinct polynomial term. This construct is then used to obtain lower bounds in the context of a proposed branch and bound scheme, which is proven to converge to a global optimal solution. A numerical example is presented to illustrate the proposed algorithm.

Spectrum Sharing for Multi-Hop Networking with Cognitive Radios

Cognitive radio (CR) capitalizes advances in signal processing and radio technology and is capable of reconfiguring RF and switching to desired frequency bands. It is a frequency-agile data communication device that is vastly more powerful than recently proposed multi-channel multi-radio (MC-MR) technology. In this paper, we investigate the important problem of multi-hop networking with CR nodes. For such a network, each node has a pool of frequency bands (typically of unequal size) that can be used for communication. The potential difference in the bandwidth among the available frequency bands prompts the need to further divide these bands into sub-bands for optimal spectrum sharing. We characterize the behavior and constraints for such a multi-hop CR network from multiple layers, including modeling of spectrum sharing and sub-band division, scheduling and interference constraints, and flow routing. We develop a mathematical formulation with the objective of minimizing the required network-wide radio spectrum resource for a set of user sessions. Since the formulated model is a mixed-integer non-linear program (MINLP), which is NP-hard in general, we develop a lower bound for the objective by relaxing the integer variables and using a linearization technique. Subsequently, we design a near-optimal algorithm to solve this MINLP problem. This algorithm is based on a novel sequential fixing procedure, where the integer variables are determined iteratively via a sequence of linear programs. Simulation results show that solutions obtained by this algorithm are very close to the lower bounds obtained via the proposed relaxation, thus suggesting that the solution produced by the algorithm is near-optimal.

A hierarchy of relaxations and convex hull characterizations for mixed-integer zero-one programming problems

This paper is concerned with the generation of tight equivalent representations for mixedinteger zero-one programming problems. For the linear case, we propose a technique which first converts the problem into a nonlinear, polynomial mixed-integer zero-one problem by multiplying the constraints with some suitable d-degree polynomial factors involving the n binary variables, for any given d E (0, . . . , n}, and subsequently linearizes the resulting problem through appropriate variable transformations. As d varies from zero to n, we obtain a hierarchy of relaxations spanning from the ordinary linear programming relaxation to the convex hull of feasible solutions. The facets of the convex hull of feasible solutions in terms of the original problem variables are available through a standard projection operation. We also suggest an alternate scheme for applying this technique which gives a similar hierarchy of relaxations, but involving fewer “complicating” constraints. Techniques for tightening intermediate level relaxations, and insights and interpretations within a disjunctive programming framework are also presented. The methodology readily extends to multilinear mixed-integer zero-one polynomial programming problems in which the continuous variables appear linearly in the problem.

Making sensor networks immortal: an energy-renewal approach with wireless power transfer

Wireless sensor networks are constrained by limited battery energy. Thus, finite network lifetime is widely regarded as a fundamental performance bottleneck. Recent breakthrough in the area of wireless power transfer offers the potential of removing this performance bottleneck, i.e., allowing a sensor network to remain operational forever. In this paper, we investigate the operation of a sensor network under this new enabling energy transfer technology. We consider the scenario of a mobile charging vehicle periodically traveling inside the sensor network and charging each sensor nodes battery wirelessly. We introduce the concept of renewable energy cycle and offer both necessary and sufficient conditions. We study an optimization problem, with the objective of maximizing the ratio of the wireless charging vehicle (WCV)s vacation time over the cycle time. For this problem, we prove that the optimal traveling path for the WCV is the shortest Hamiltonian cycle and provide a number of important properties. Subsequently, we develop a near-optimal solution by a piecewise linear approximation technique and prove its performance guarantee.

On renewable sensor networks with wireless energy transfer

Traditional wireless sensor networks are constrained by limited battery energy. Thus, finite network lifetime is widely regarded as a fundamental performance bottleneck. Recent breakthrough in the area of wireless energy transfer offers the potential of removing such performance bottleneck, i.e., allowing a sensor network remain operational forever. In this paper, we investigate the operation of a sensor network under this new enabling energy transfer technology. We consider the scenario of a mobile charging vehicle periodically traveling inside the sensor network and charging each sensor nodes battery wirelessly. We introduce the concept of renewable energy cycle and offer both necessary and sufficient conditions. We study an optimization problem, with the objective of maximizing the ratio of the wireless charging vehicle (WCV)s vacation time over the cycle time. For this problem, we prove that the optimal traveling path for the WCV is the shortest Hamiltonian cycle and provide a number of important properties. Subsequently, we develop a near-optimal solution and prove its performance guarantee.

A mathematical programming approach for determining oligopolistic market equilibrium

During the past several years it has become increasingly common to use mathematical programming methods for deriving economic equilibria of supply and demand. Well-defined approaches exist for the case of a single firm (monopoly) and for the case of many firms (perfect competition). In this paper a certain family of convex programs is formulated to determine equilibria for the case of a few firms (oligopoly). Solutions to this family of convex programs are shown to be Nash equilibria in the formal sense ofN person games. This equivalence leads to a mathematical programming-based algorithm for determining an oligopolistic market equilibrium.

Optimal Spectrum Sharing for Multi-Hop Software Defined Radio Networks

Software defined radio (SDR) capitalizes advances in signal processing and radio technology and is capable of reconfiguring RF and switching to desired frequency bands. It is a frequency-agile data communication device that is vastly more powerful than recently proposed multi-channel multi-radio (MC-MR) technology. In this paper, we investigate the important problem of multi-hop networking with SDR nodes. For such network, each node has a pool of frequency bands (not necessarily of equal size) that can be used for communication. The uneven size of bands in the radio spectrum prompts the need of further division into sub-bands for optimal spectrum sharing. We characterize behaviors and constraints for such multi-hop SDR network from multiple layers, including modeling of spectrum sharing and sub-band division, scheduling and interference constraints, and flow routing. We give a formal mathematical formulation with the objective of minimizing the required network-wide radio spectrum resource for a set of user sessions. Since such problem formulation falls into mixed integer non-linear programming (MINLP), which is NP-hard in general, we develop a lower bound for the objective by relaxing the integer variables and linearization. Subsequently, we develop a near-optimal algorithm to this MINLP problem. This algorithm is based on a novel sequential fixing procedure, where the integer variables are determined iteratively via a sequence of linear programming. Simulation results show that solutions obtained by this algorithm are very close to lower bounds obtained via relaxation, thus suggesting that the solution produced by the algorithm is near-optimal.