Characterizing hidden convexity of quadratic systems: when is the joint image $\mathrm{U}(f,g_1,\ldots,g_m)$ convex for $m\ge 2$?

For quadratic functions $f,g_1,\ldots,g_m$ on $\mathbb{R}^n$, the set $\mathrm{U}(f,g_1,\ldots,g_m)=\{(f(x),g_1(x),\ldots,g_m(x)):x\in\mathbb{R}^n\}+\mathbb{R}^{m+1}_+$ is the mechanism behind hidden convexity in nonconvex quadratic programming: when this set is convex and Slater's condition holds, the generalized S-lemma with $m$ quadratic constraints is valid, which yields strong Lagrangian duality, exact semidefinite relaxations, and necessary-and-sufficient global optimality conditions for quadratically constrained quadratic programs. For a single quadratic constraint ($m=1$) the set is always convex, but for $m\ge 2$ it can fail to be convex. The source paper (Huy, Hung, Nghi, Tuan, Tuyen, 2026) explicitly poses the question of under what assumptions $\mathrm{U}(f,g_1,\ldots,g_m)$ is convex and gives only a partial answer through sufficient conditions (H1)–(H2) and trust-region conditions (H3)–(H4). No verifiable characterization is known for $m\ge 2$: the open core is to determine the assumptions — ideally necessary and sufficient, and in any case covering regimes beyond the known dimension conditions, Z-matrix hypotheses, and (H1)–(H4) — under which the joint image plus nonnegative orthant of a quadratic system is convex.

Background

A quadratically constrained quadratic program (QCQP) minimizes a quadratic function $f(x)=x^T Ax+a^T x+\alpha$ subject to quadratic inequalities $g_i(x)=x^T B_i x+b_i^T x+\beta_i\le 0$, $i=1,\ldots,m$, where $A,B_i$ are real symmetric $n\times n$ matrices. QCQPs are generally nonconvex, yet several fundamental instances behave as if convex — a phenomenon called hidden convexity. Its archetype is the S-lemma (S-procedure) of Yakubovich: for two quadratic functions $f,g$ with a Slater point (a point $\bar x$ with $g(\bar x)<0$), the implication $(\forall x,\ g(x)\le 0\Rightarrow f(x)\le 0)$ holds if and only if $f-\lambda g\le 0$ on all of $\mathbb{R}^n$ for some $\lambda\ge 0$. The S-lemma is equivalent to zero Lagrangian duality gap for QCQPs with one quadratic constraint, and it underlies exact semidefinite programming (SDP) relaxations of the trust-region subproblem (minimizing a quadratic over a Euclidean ball); the survey by Pólik and Terlaky (2007) recounts this theory and records that the lemma fails in general once several quadratic constraints are present.

The geometric engine of the S-lemma is the joint image of the quadratic system augmented by the nonnegative orthant, $\mathrm{U}(f,g_1,\ldots,g_m):=\{(f(x),g_1(x),\ldots,g_m(x)):x\in\mathbb{R}^n\}+\mathbb{R}^{m+1}_{+}.$ If $\mathrm{U}$ is convex and Slater's condition holds, separating a point from the convex set $\mathrm{U}$ produces the multipliers of the generalized S-lemma with $m$ constraints, which in turn gives strong duality, exact Lagrangian and SDP relaxations, and global optimality conditions for the corresponding QCQP. Convexity of $\mathrm{U}$ is also equivalent to the map $x\mapsto(f(x),g_1(x),\ldots,g_m(x))$ being convexlike (meaning that for any $x_1,x_2\in\mathbb{R}^n$ and any $\theta\in[0,1]$ there exists $x_3\in\mathbb{R}^n$ with $(f(x_3),g_1(x_3),\ldots,g_m(x_3))\le\theta(f(x_1),g_1(x_1),\ldots,g_m(x_1))+(1-\theta)(f(x_2),g_1(x_2),\ldots,g_m(x_2))$ componentwise — equivalently, the image of the map plus the nonnegative orthant is convex), a property used in minimax theory and multiobjective quadratic optimization, where the same image-plus-orthant object is studied under the name of the upper image and convexlikeness (Eichfelder, Kugelmann and Yıldırım 2026).

For one quadratic constraint ($m=1$, two quadratic functions) the question is completely settled: Dines (1941) proved convexity of the joint image of two homogeneous quadratic forms; Polyak (1998) proved convexity of $\mathrm{U}(f,g)$ when $g$ is strictly convex and connected it to control; Tuy and Tuan (2013) proved that $\mathrm{U}(f,g)$ is convex for two arbitrary quadratic functions on an affine manifold (a complete proof is given in Huy et al. (2026) and in the literature on range sets of two quadratic functions, e.g., Nguyen, Chu and Sheu 2022, extended to Hilbert spaces by Nguyen 2025; an alternative proof in the multiobjective framework is given by Eichfelder, Kugelmann and Yıldırım 2026).

For $m\ge 2$ (at least two quadratic inequality constraints, some of which may be affine) the situation changes: Jeyakumar and Li (2014, Example 2.1) exhibited a system where $\mathrm{U}(f,g_1,g_2)$ is not convex, so hidden convexity fails in general — a fact echoed by De Lara and Hiriart-Urruty (2022), who conjectured nonconvexity for $m\ge 2$. All known positive results are sufficient conditions with strong structural hypotheses. Jeyakumar, Lee and Li (2009) assume the associated matrices are Z-matrices (symmetric matrices whose off-diagonal entries are all nonpositive). Jeyakumar and Li (2014) assume $g_1$ strictly convex, $g_2,\ldots,g_m$ affine, and a dimension condition $\dim\ker(A-\bar\lambda I_n)\ge s+1$, where $\bar\lambda$ is chosen with $A-\bar\lambda I_n\succeq 0$ and $s=\dim\operatorname{span}\{b_2,\ldots,b_m\}$ ($\ker$ denotes the null space and $I_n$ the identity matrix). Bomze, Jeyakumar and Li (2018) treat the Celis–Dennis–Tapia (CDT, two-ball trust-region) problem with an extended dimension condition. Huy et al. (2026) prove, in the standing setting of convex quadratic constraints $B_i\succeq 0$ with at least one $B_{i_0}\succ 0$, that $\mathrm{U}$ is convex under: (H1) existence of $\lambda$ with $A+\lambda B_{i_0}\succeq 0$ for some strictly convex constraint $g_{i_0}$, and (H2) nontriviality of the recession cone (the set of directions $v$ along which a closed convex set recedes to infinity) of the system $\{x: f(x)+\lambda g_{i_0}(x)\le 0,\ g_i(x)\le 0,\ i\ne i_0\}$; specialized versions (H3)–(H4) cover the trust-region problem with linear inequality constraints $\inf\{x^T Ax+a^T x:\ h_1(x)\le 0,\ h_i(x)\le 0,\ i=2,\ldots,m\}$, where $h_1(x)=\|x-x_0\|^2-\alpha$ with $\alpha>0$ and $h_i(x)=b_i^T x-\beta_i$ are affine, and are shown to be strictly weaker than the Jeyakumar–Li dimension condition: with $\bar\lambda=\lambda_{\min}(A)$, condition (H3) is the existence of a nonzero $v$ with $Av-\bar\lambda v=0$, $(a+2\bar\lambda x_0)^T v\le 0$ and $b_i^T v\le 0$ for all $i\ge 2$, and condition (H4) is the weaker kernel condition $\ker(A-\bar\lambda I_n)\cap\bigcap_{i\ge 2}b_i^\perp\neq\{0\}$ (so (H4) implies (H3)). Related but distinct lines study convexity of the joint numerical range (the image of the unit sphere under quadratic forms — Dines, Polyak, and extensions by Song and Xia 2021) and sufficient conditions for convexity of images of multidimensional quadratic maps (Dymarsky 2014); these concern different sets and do not characterize $\mathrm{U}$.

What no available result provides is a verifiable characterization — necessary and sufficient conditions, or sufficient conditions substantially broader than the above — for convexity of $\mathrm{U}(f,g_1,\ldots,g_m)$ when $m\ge 2$. Huy et al. (2026) state this explicitly as an open question to which they give only a partial answer.

Problem Statement

Let $f(x)=x^T Ax+a^T x+\alpha$ and $g_i(x)=x^T B_i x+b_i^T x+\beta_i$, $i=1,\ldots,m$, be real quadratic functions on $\mathbb{R}^n$ (in the standard setting the constraint Hessians satisfy $B_i\succeq 0$, with at least one $B_{i_0}\succ 0$), and define $\mathrm{U}(f,g_1,\ldots,g_m):=\{(f(x),g_1(x),\ldots,g_m(x)):x\in\mathbb{R}^n\}+\mathbb{R}^{m+1}_{+}.$ For systems with $m\ge 2$ quadratic constraints, where hidden convexity is known to fail in general (Jeyakumar–Li 2014, Example 2.1), the open problem posed in the source paper is: under what assumptions on $(f,g_1,\ldots,g_m)$ is $\mathrm{U}(f,g_1,\ldots,g_m)$ convex?

The target is a verifiable characterization — ideally necessary and sufficient — expressed in terms of the data (the Hessians $A,B_i$, the linear terms $a,b_i$, and the constants), holding exactly when $\mathrm{U}$ is convex. The known sufficient conditions — the (H1)–(H2) conditions and trust-region conditions (H3)–(H4) of Huy et al. (2026), the dimension condition of Jeyakumar–Li (2014) and its extension by Bomze–Jeyakumar–Li (2018), and the Z-matrix conditions of Jeyakumar–Lee–Li (2009) — leave regimes not covered by any of them. Any correct characterization reproduces the known instances: the convex cases of Huy et al. (2026, Examples 3.4–3.5), which evade the earlier dimension conditions, and the nonconvex cases (Jeyakumar–Li 2014, Example 2.1; Huy et al. 2026, Example 3.6). The scientific stake is direct: convexity of $\mathrm{U}$ together with Slater's condition is equivalent to the validity of the generalized S-lemma with $m$ quadratic constraints, and hence to strong Lagrangian duality, exact semidefinite relaxation, and necessary-and-sufficient global optimality conditions for the corresponding nonconvex QCQP.

References

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