In this paper we introduce a new parameterized Quadratic Decision Rule (QDR), a generalisation of the commonly employed Affine Decision Rule (ADR), for two-stage linear adjustable robust optimization problems with ellipsoidal uncertainty and show that (affinely parameterized) linear adjustable robust optimization problems with QDRs are numerically tractable by presenting exact semi-definite program (SDP) and second order cone program (SOCP) reformulations. Under these QDRs, we also establish that exact conic program reformulations also hold for two-stage linear ARO problems, containing also adjustable variables in their objective functions. We then show via numerical experiments on lot-sizing problems with uncertain demand that adjustable robust linear optimization problems with QDRs improve upon the ADRs in their performance both in the worst-case sense and after simulated realization of the uncertain demand relative to the true solution.
We present a geometrical analysis on the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems with a class of geometrically defined nonconvex conic programs and their covexification. The class of nonconvex conic programs is described with a linear objective functionin a linear space $V$, and the constraint set is represented geometrically as the intersection of a nonconvex cone $K \subset V$, a face $J$ of the convex hull of $K$ and a parallel translation $L$ of a supporting hyperplane of the nonconvex cone $K$. …
In an effort to develop an alternative approach to traditional sparse reformulations, we will provide a new type of convex reformulation of a large class of stochastic quadratically constrained quadratic optimization problems that is similar to Burer's reformulation, but lifts the variables into a comparatively lower dimensional space. The reformulation rests on a generalization of the set-completely positive matrix cone. This cone can then be approximated via inner and outer approximations in order to obtain upper and lower bounds, which potentially close the optimality gap, and hence can giv…
Aggregation of heating, ventilation, and air conditioning (HVAC) loads can provide reserves to absorb volatile renewable energy, especially solar photo-voltaic (PV) generation. In this paper, we decide HVAC control schedules under uncertain PV generation, using a distributionally robust chance-constrained (DRCC) building load control model under two typical ambiguity sets: the moment-based and Wasserstein ambiguity sets. We derive mixed-integer linear programming (MILP) reformulations for DRCC problems under both sets. Especially, for the Wasserstein ambiguity set, we utilize the right-hand si…
In this paper, we show that the popular K-means clustering problem can equivalently be reformulated as a conic program of polynomial size. The arising convex optimization problem is NP-hard, but amenable to a tractable semidefinite programming (SDP) relaxation that is tighter than the current SDP relaxation schemes in the literature. In contrast to the existing schemes, our proposed SDP formulation gives rise to solutions that can be leveraged to identify the clusters. We devise a new approximation algorithm for K-means clustering that utilizes the improved formulation and empirically illustra…
We present algorithm MIQCR-CB that is an advancement of method MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving mixed-integer quadratic programs and works in two phases: the first phase determines an equivalent quadratic formulation with a convex objective function by solving a semidefinite problem $(SDP)$, and, in the second phase, the equivalent formulation is solved by a standard solver. As the reformulation relies on the solution of a large-scale semidefinite program, it is not tractable by existing semidefinite solvers, already for medium sized problems. To sur…
The concept of anchored solutions is proposed as a new robust optimization approach to the Resource-Constrained Project Scheduling Problem (RCPSP) under processing times uncertainty. The Anchor-Robust RCPSP is defined, to compute a baseline schedule with bounded makespan, sequencing decisions, and a max-size subset of jobs with guaranteed starting times, called anchored set. It is shown that the Adjustable-Robust RCPSP from the literature fits within the framework of anchored solutions. The Anchor-Robust RCPSP and the Adjustable-Robust RCPSP can benefit from each other to find both a worst-cas…
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 (…
In this paper, using an optimal partition approach, we study the parametric analysis of a second-order conic optimization problem, where the objective function is perturbed along a fixed direction. We characterize the notions of so-called invariancy set and nonlinearity interval, which serve as stability regions of the optimal partition. We then propose, under the strict complementarity condition, an iterative procedure to compute a nonlinearity interval of the optimal partition. Furthermore, under primal and dual nondegeneracy conditions, we show that a boundary point of a nonlinearity interv…
This work proposes a framework for multistage adjustable robust optimization that unifies the treatment of three different types of endogenous uncertainty, where decisions, respectively, (i) alter the uncertainty set, (ii) affect the materialization of uncertain parameters, and (iii) determine the time when the true values of uncertain parameters are observed. We provide a systematic analysis of the different types of endogenous uncertainty and highlight the connection between optimization under endogenous uncertainty and active learning. We consider decision-dependent polyhedral uncertainty s…
We provide an exact deterministic reformulation for data-driven chance constrained programs over Wasserstein balls. For individual chance constraints as well as joint chance constraints with right-hand side uncertainty, our reformulation amounts to a mixed-integer conic program. In the special case of a Wasserstein ball with the $1$-norm or the $\infty$-norm, the cone is the nonnegative orthant, and the chance constrained program can be reformulated as a mixed-integer linear program. Our reformulation compares favourably to several state-of-the-art data-driven optimization schemes in our numer…
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 (…
We study decision dependent distributionally robust optimization models, where the ambiguity sets of probability distributions can depend on the decision variables. These models arise in situations with endogenous uncertainty. The developed framework includes two-stage decision dependent distributionally robust stochastic programming as a special case. Decision dependent generalizations of five types of ambiguity sets are considered. These sets are based on bounds on moments, Wasserstein metric, $ϕ$-divergence and Kolmogorov-Smirnov test. For the finite support case, we use linear, conic or La…
The classical alternating current optimal power flow problem is highly nonconvex and generally hard to solve. Convex relaxations, in particular semidefinite, second-order cone, convex quadratic, and linear relaxations, have recently attracted significant interest. The semidefinite relaxation is the strongest among them and is exact for many cases. However, the computational efficiency for solving large-scale semidefinite optimization is lower than for second-order cone optimization. We propose a conic relaxation obtained by combining semidefinite optimization with the reformulation-linearizati…
We address high-dimensional zero-one random parameters in two-stage convex conic optimization problems. Such parameters typically represent failures of network elements and constitute rare, high-impact random events in several applications. Given a sparse training dataset of the parameters, we motivate and study a distributionally robust formulation of the problem using a Wasserstein ambiguity set centered at the empirical distribution. We present a simple, tractable, and conservative approximation of this problem that can be efficiently computed and iteratively improved. Our method relies on …
We present a unifying framework for generating extended formulations for the polyhedral outer approximations used in algorithms for mixed-integer convex programming (MICP). Extended formulations lead to fewer iterations of outer approximation algorithms and generally faster solution times. First, we observe that all MICP instances from the MINLPLIB2 benchmark library are conic representable with standard symmetric and nonsymmetric cones. Conic reformulations are shown to be effective extended formulations themselves because they encode separability structure. For mixed-integer conic-representa…
In this paper, we study a class of fractional semi-infinite polynomial programming problems involving s.o.s-convex polynomial functions. For such a problem, by a conic reformulation proposed in our previous work and the quadratic modules associated with the index set, a hierarchy of semidefinite programming (SDP) relaxations can be constructed and convergent upper bounds of the optimum can be obtained. In this paper, by introducing Lasserre's measure-based representation of nonnegative polynomials on the index set to the conic reformulation, we present a new SDP relaxation method for the consi…
The advances in conic optimization have led to its increased utilization for modeling data uncertainty. In particular, conic mean-risk optimization gained prominence in probabilistic and robust optimization. Whereas the corresponding conic models are solved efficiently over convex sets, their discrete counterparts are intractable. In this paper, we give a highly effective successive quadratic upper-bounding procedure for discrete mean-risk minimization problems. The procedure is based on a reformulation of the mean-risk problem through the perspective of its convex quadratic term. Computationa…
This paper considers the resource-constrained project scheduling problem with uncertain activity durations. We assume that activity durations lie in a budgeted uncertainty set, and follow a robust two-stage approach, where a decision maker must resolve resource conflicts subject to the problem uncertainty, but can determine activity start times after the uncertain activity durations become known. We introduce a new reformulation of the second-stage problem, which enables us to derive a compact robust counterpart to the full two-stage adjustable robust optimisation problem. Computational experi…
We consider minimizing a conic quadratic objective over a polyhedron. Such problems arise in parametric value-at-risk minimization, portfolio optimization, and robust optimization with ellipsoidal objective uncertainty; and they can be solved by polynomial interior point algorithms for conic quadratic optimization. However, interior point algorithms are not well-suited for branch-and-bound algorithms for the discrete counterparts of these problems due to the lack of effective warm starts necessary for the efficient solution of convex relaxations repeatedly at the nodes of the search tree. In o…
This paper introduces ROmodel, an open source Python package extending the modeling capabilities of the algebraic modeling language Pyomo to robust optimization problems. ROmodel helps practitioners transition from deterministic to robust optimization through modeling objects which allow formulating robust models in close analogy to their mathematical formulation. ROmodel contains a library of commonly used uncertainty sets which can be generated using their matrix representations, but it also allows users to define custom uncertainty sets using Pyomo constraints. ROmodel supports adjustable v…
In this paper we present a complete iteration complexity analysis of inexact first order Lagrangian and penalty methods for solving cone constrained convex problems that have or may not have optimal Lagrange multipliers that close the duality gap. We first assume the existence of optimal Lagrange multipliers and study primal-dual first order methods based on inexact information and augmented Lagrangian smoothing or Nesterov type smoothing. For inexact (fast) gradient augmented Lagrangian methods we derive a total computational complexity of $\mathcal{O}\left(\frac{1}ε\right)$ projections onto …
Alternating current optimal power flow (AC OPF) is one of the most fundamental optimization problems in electrical power systems. It can be formulated as a semidefinite program (SDP) with rank constraints. Solving AC OPF, that is, obtaining near optimal primal solutions as well as high quality dual bounds for this non-convex program, presents a major computational challenge to today's power industry for the real-time operation of large-scale power grids. In this paper, we propose a new technique for reformulation of the rank constraints using both principal and non-principal 2-by-2 minors of t…
We propose a homogeneous primal-dual interior-point method to solve sum-of-squares optimization problems by combining non-symmetric conic optimization techniques and polynomial interpolation. The approach optimizes directly over the sum-of-squares cone and its dual, circumventing the semidefinite programming (SDP) reformulation which requires a large number of auxiliary variables. As a result, it has substantially lower theoretical time and space complexity than the conventional SDP-based approach. Although our approach avoids the semidefinite programming reformulation, an optimal solution to …
This note establishes a limiting formula for the conic Lagrangian dual of a convex infinite optimization problem, correcting the classical version of Karney [Math. Programming 27 (1983) 75-82] for convex semi-infinite programs. A reformulation of the convex infinite optimization problem with a single constraint leads to a limiting formula for the corresponding Lagrangian dual, called sup-dual, and also for the primal problem in the case when strong Slater condition holds, which also entails strong sup-duality.
To address the challenge of the renewable energy uncertainty, the ISO New England (ISO-NE) has proposed to apply do-not-exceed (DNE) limits, which represent the maximum nodal injection of renewable energy the grid can accommodate. Unfortunately, it appears challenging to compute DNE limits that simultaneously maintain the system flexibility and incorporate a large portion of the available renewable energy at the minimum cost. In addition, it is often challenging to accurately estimate the joint probability distribution of the renewable energy. In this paper, we propose a two-stage distribution…
We study the trust-region subproblem (TRS) of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints. Despite having a nonconvex objective, it is known that the classical TRS and a number of its variants are polynomial-time solvable. In this paper, we follow a second-order cone (SOC) based approach to derive an exact convex reformulation of the TRS under a structural condition on the conic constraint. Our structural condition is immediately satisfied when there is no additional conic constraints, and it generalizes several such conditions studied in the …
We study quadratic optimization with indicator variables and an M-matrix, i.e., a PSD matrix with non-positive off-diagonal entries, which arises directly in image segmentation and portfolio optimization with transaction costs, as well as a substructure of general quadratic optimization problems. We prove, under mild assumptions, that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem. To strengthen the formulation, we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variabl…
In this paper we present BilevelJuMP, a new Julia package to support bilevel optimization within the JuMP framework. The package is a Julia library that enables the user to describe both upper and lower-level optimization problems using the JuMP algebraic syntax. Due to the generality and flexibility our library inherits from JuMP's syntax, our package allows users to model bilevel optimization problems with conic constraints in the lower level and all JuMP supported constraints in the upper level (Conic, Quadratic, Non-Linear, Integer, etc.). Moreover, the user-defined problem can be subseque…
In a common formulation of semi-infinite programs, the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval. If this function is sufficiently closely approximable by a polynomial or a rational function, then the semi-infinite program can be reformulated as an equivalent semidefinite program. Solving this semidefinite program is challenging if the polynomials involved are of high degree, due to numerical difficulties and bad scaling arising both from the polynomial approximations and from the fact that the semidefinite p…
We derive computationally tractable formulations of the robust counterparts of convex quadratic and conic quadratic constraints that are concave in matrix-valued uncertain parameters. We do this for a broad range of uncertainty sets. In particular, we show how to reformulate the support functions of uncertainty sets represented in terms of matrix norms and cones. Our results provide extensions to known results from the literature. We also consider hard quadratic constraints; those that are convex in uncertain matrix-valued parameters. For the robust counterpart of such constraints we derive in…
