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We introduce and study conic geometric programs (CGPs), which are convex optimization problems that unify geometric programs (GPs) and conic optimization problems such as semidefinite programs (SDPs). A CGP consists of a linear objective function that is to be minimized subject to affine constraints, convex conic constraints, and upper bound constraints on sums of exponential and affine functions. The conic constraints are the central feature of conic programs such as SDPs, while upper bounds on combined exponential/affine functions are generalizations of the types of constraints found in GPs. The dual of a CGP involves the maximization of the negative relative entropy between two nonnegative vectors jointly, subject to affine and conic constraints on the two vectors. Although CGPs contain GPs and SDPs as special instances, computing global optima of CGPs is not much harder than solving GPs and SDPs. More broadly, the CGP framework facilitates a range of new applications that fall outside the scope of SDPs and GPs. Specifically, we demonstrate the utility of CGPs in providing solutions to problems such as permanent maximization, hitting-time estimation in dynamical systems, the computation of the capacity of channels transmitting quantum information, and robust optimization formulations of GPs.
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We introduce log-log convex programs, which are optimization problems with positive variables that become convex when the variables, objective functions, and constraint functions are replaced with their logs, which we refer to as a log-log transformation. This class of problems generalizes traditional geometric programming and generalized geometric programming, and it includes interesting problems involving nonnegative matrices. We give examples of log-log convex functions, some well-known and some less so, and we develop an analog of disciplined convex programming, which we call disciplined geometric programming. Disciplined geometric programming is a subclass of log-log convex programming generated by a composition rule and a set of functions with known curvature under the log-log transformation. Finally, we describe an implementation of disciplined geometric programming as a reduction in CVXPY 1.0.
Shunt FACTS devices, such as, a Static Var Compensator (SVC), are capable of providing local reactive power compensation. They are widely used in the network to reduce the real power loss and improve the voltage profile. This paper proposes a planning model based on mixed integer conic programming (MICP) to optimally allocate SVCs in the transmission network considering load uncertainty. The load uncertainties are represented by a number of scenarios. Reformulation and linearization techniques are utilized to transform the original non-convex model into a convex second order cone programming (SOCP) model. Numerical case studies based on the IEEE 30-bus system demonstrate the effectiveness of the proposed planning model.
In this paper we consider a class of convex conic programming. In particular, we propose an inexact augmented Lagrangian (I-AL) method for solving this problem, in which the augmented Lagrangian subproblems are solved approximately by a variant of Nesterovs optimal first-order method. We show that the total number of first-order iterations of the proposed I-AL method for computing an $epsilon$-KKT solution is at most $mathcal{O}(epsilon^{-7/4})$. We also propose a modified I-AL method and show that it has an improved iteration-complexity $mathcal{O}(epsilon^{-1}logepsilon^{-1})$, which is so far the lowest complexity bound among all first-order I-AL type of methods for computing an $epsilon$-KKT solution. Our complexity analysis of the I-AL methods is mainly based on an analysis on inexact proximal point algorithm (PPA) and the link between the I-AL methods and inexact PPA. It is substantially different from the existing complexity analyses of the first-order I-AL methods in the literature, which typically regard the I-AL methods as an inexact dual gradient method. Compared to the mostly related I-AL methods cite{Lan16}, our modified I-AL method is more practically efficient and also applicable to a broader class of problems.
Adaptive robust optimization problems are usually solved approximately by restricting the adaptive decisions to simple parametric decision rules. However, the corresponding approximation error can be substantial. In this paper we show that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions. Specifically, when the ambiguity set constitutes a 2-Wasserstein ball centered at a discrete distribution, then the distributionally robust linear program is equivalent to a copositive program (if the problem has complete recourse) or can be approximated arbitrarily closely by a sequence of copositive programs (if the problem has sufficiently expensive recourse). These results directly extend to the classical robust setting and motivate strong tractable approximations of two-stage problems based on semidefinite approximations of the copositive cone. We also demonstrate that the two-stage distributionally robust optimization problem is equivalent to a tractable linear program when the ambiguity set constitutes a 1-Wasserstein ball centered at a discrete distribution and there are no support constraints.
We provide a condition-based analysis of two interior-point methods for unconstrained geometric programs, a class of convex programs that arise naturally in applications including matrix scaling, matrix balancing, and entropy maximization. Our condition numbers are natural geometric quantities associated with the Newton polytope of the geometric program, and lead to diameter bounds on approximate minimizers. We also provide effective bounds on the condition numbers both in general and under combinatorial assumptions on the Newton polytope. In this way, we generalize the iteration complexity of recent interior-point methods for matrix scaling and matrix balancing. Recently, there has been much work on algorithms for certain optimization problems on Lie groups, known as capacity and scaling problems. For commutative groups, these problems reduce to unconstrained geometric programs, which serves as a particular source of motivation for our work.
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